Systems and methods for generating a design for a gliding board

ABSTRACT

Systems and methods are provided for generating a design for a gliding board. The method involves operating a processor to: define a desired carved turn of the gliding board; define a desired global curvature profile; generate a desired deformed shape of the gliding board during the desired carved turn; generate a sidecut profile of the gliding board; generate a width profile of the gliding board; generate a camber profile of the gliding board; generate at least one stiffness design variable profile; generate a total load profile; modify at least the width profile, the sidecut profile and at least one of the at least one stiffness design variable profile at least once; and define the design for the gliding board based at least on the width profile, the camber profile, and the at least one stiffness design variable profile.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims the benefit of U.S. Provisional Application No.62/787,443 filed on Jan. 2, 2019, which is incorporated by referenceherein in its entirety.

FIELD

The described embodiments relate to systems and methods for generating adesign for a gliding board.

BACKGROUND

Skiing is an activity that has broad appeal. Skis have been developedfor use in a variety of skiing activities. Examples include downhillskis and cross-country skis.

Generally, athletes use skis which have been roughly sized andconfigured according to the environment and conditions in which thoseskis are to be used. For example, a larger athlete will typically uselarger skis than will a smaller athlete.

However, although skis customized for the athlete and conditions offermany performance benefits, many athletes continue to use skis which are,at best, roughly chosen to correspond to conditions of use.

In the context of alpine (downhill) skiing, a carved turn can becharacterized as a turn that is executed in a manner that minimizes thequantity of snow that is displaced from its resting position on theground, thus minimizing the drag (snow resistance) that is imposed uponthe ski during the turn. The term “carving” is often used in referenceto the act of carrying out carved turns while skiing.

A broad variety of alpine skiing activities exist, and a correspondinglybroad variety of alpine skis has been developed in order to accommodateeach type of alpine skiing activity. For example, athletes wishing toski on freshly fallen unprepared virgin snow (powder snow) would likelychoose to use skis that are relatively wide, thus offering a large totalsurface area that would help the skis to float near the surface of thesnow. Conversely, athletes wishing to ski on prepared pistes comprisingsmooth surfaces of firm snow (groomed trails or race courses) wouldlikely choose to use skis that are relatively narrow, and featureprecisely engineered flexural stiffness profiles that help to maximizethe engagement of the edges of the skis with the snow, thus maximizinggrip and minimizing drag during carved turns.

When a ski interacts with a soft substrate (such as a thick layer ofpowder snow), any small discrepancies in the deflected shape of the skiwould have a relatively minute influence upon the overall snow pressuredistribution that acts upon the base surface of the ski. Conversely,when a ski interacts with a firm substrate (such as a groomed alpinetrail), any small discrepancies in the deflected shape of the ski couldhave a marked influence upon the snow pressure distribution that actsupon the base surface of the ski. Consequently, alpine skis that areintended for on-piste carving are particularly sensitive to therelationship that exists between the various geometric and mechanicalspecifications of the ski design, as well as the precision and accuracywith which said skis were manufactured.

Existing approaches to ski design attempt to account for the complexgeometric and mechanical aspects of skis, and the interaction betweenthose skis and various types of snow conditions, largely based on designexperience and/or engineering judgement, and often in the context ofextremely simplified models of the skis and snow. There exists a needfor an automated approach to alpine ski design that integrates acomprehensive model of the skis and the snow on which they will be used.

SUMMARY

The various embodiments described herein generally relate to systems andmethods for generating a design of a ski.

In accordance with some embodiments, there is provided acomputer-implemented method for generating a design for a gliding board.The method involves operating a processor to: define a desired carvedturn of the gliding board, the desired carved turn being defined atleast by a nominal edging angle and an athlete load profile, wherein theathlete load profile represents a load that is applied by an athlete tothe gliding board during the desired carved turn; define a desiredglobal curvature profile, wherein the desired global curvature profilecorresponds to a desired snow trace profile for the desired carved turn;generate a desired deformed shape of the gliding board during thedesired carved turn, the desired deformed shape of the gliding boardbeing defined at least by a desired total curvature profile, wherein thedesired total curvature profile is initially set to correspond to thedesired global curvature profile; generate a sidecut profile of thegliding board; generate a width profile of the gliding board based atleast on the sidecut profile; generate a camber profile of the glidingboard; generate at least one stiffness design variable profile, whereinthe at least one stiffness design variable profile, in conjunction withat least the width profile and at least one gliding board materialproperty, dictates a resulting flexural stiffness profile and aresulting torsional stiffness profile of the gliding board; generate atotal load profile based at least on the athlete load profile, whereinthe total load profile represents a total load that is applied to thegliding board during the desired carved turn, and wherein generating thetotal load profile involves generating a desired snow penetration depthprofile; modify at least the width profile, the sidecut profile and atleast one of the at least one stiffness design variable profile at leastonce; and define the design for the gliding board based at least on thewidth profile, the camber profile, and the at least one stiffness designvariable profile. The processor can modify at least the width profile,the sidecut profile and at least one of the at least one stiffnessdesign variable profile at least once by: calculating a desired flexuralstiffness profile of the gliding board based at least on the total loadprofile, such that the desired flexural stiffness profile approximatelyachieves the desired total curvature profile during the carved turn;modifying at least one of the at least one stiffness design variableprofile such that the resulting flexural stiffness profile isapproximately equal to the desired flexural stiffness profile;calculating a torsional deformation profile of the gliding board duringthe carved turn based at least on the total load profile; modifying thedesired total curvature profile based at least on the torsionaldeformation profile in order to achieve a resulting global curvatureprofile that is approximately equal to the desired global curvatureprofile during the carved turn; modifying the sidecut profile based atleast on the torsional deformation profile in order to achieve aresulting snow penetration depth profile that is approximately equal tothe desired snow penetration depth profile; and modifying the widthprofile based at least on the modified sidecut profile.

In some embodiments, modifying at least the width profile, the sidecutprofile and at least one of the at least one stiffness design variableprofile at least once involves: iteratively repeating the steps ofcalculating the desired flexural stiffness profile, modifying at leastone of the at least one stiffness design variable profile, calculatingthe torsional deformation profile, modifying the desired total curvatureprofile, modifying the sidecut profile, and modifying the width profileuntil a difference between at least one of the width profile, thesidecut profile, and at least one of the at least one stiffness designvariable profile associated with a current iteration and at least one ofthe width profile, the sidecut profile, and at least one of the at leastone stiffness design variable profile associated with the previousiteration is less than a predetermined threshold.

In some embodiments, defining the desired global curvature profileinvolves: defining the desired global curvature profile to correspond toa carved turn that exhibits an approximately constant turning radius.

In some embodiments, each of the desired global curvature profile andthe resulting global curvature profile embodies a curvilinear form thatcan be determined by projecting a curvilinear geometry of acorresponding snow trace profile onto an oblique projection plane thatis inclined with respect to the plane on which the respective snow traceprofile exists.

In some embodiments, generating the sidecut profile involves: defining adesired geometry of a snow penetration depth profile, wherein the snowpenetration depth profile represents the penetration depth of thegliding board into snow during the desired carved turn; and generatingthe sidecut profile based at least on the desired global curvatureprofile, the desired geometry of the snow penetration depth profile, andthe nominal edging angle.

In some embodiments, defining the desired geometry of the snowpenetration depth profile involves: defining the desired geometry of thesnow penetration depth profile to approximately follow a linear functionover a length of an effective base region of the gliding board.

In some embodiments, generating the camber profile involves: determininga first assumed total curvature profile of the gliding board and a firstload profile of the gliding board corresponding to a first gliding boardedging angle; determining a second assumed total curvature profile ofthe gliding board and a second load profile of the gliding boardcorresponding to a second gliding board edging angle; and generating thecamber profile such that it facilitates a gliding board design that iscapable of satisfying the first assumed total curvature profile with thefirst load profile, and is also capable of satisfying the second assumedtotal curvature profile with the second load profile.

In some embodiments, at least one of the at least one stiffness designvariable profile includes a core thickness profile of the gliding board.

In some embodiments, calculating the desired flexural stiffness profileinvolves: calculating the desired flexural stiffness profile of thegliding board based at least on the camber profile, the desired totalcurvature profile, and the total load profile.

In some embodiments, modifying at least one of the at least onestiffness design variable profile involves: finding the at least onestiffness design variable profile such that the resulting flexuralstiffness profile is approximately equal to the desired flexuralstiffness profile, wherein the at least one stiffness design variableprofile includes a plurality of local stiffness design variable values,wherein each local stiffness design variable value is limited to apredetermined range of allowable values.

In some embodiments, modifying at least the width profile, the sidecutprofile and at least one of the at least one stiffness design variableprofile at least once further involves modifying the camber profile atleast once by: determining a resulting flexural stiffness profile of thegliding board based at least on the at least one stiffness designvariable profile, the width profile, and at least one gliding boardmaterial property; determining a resulting flexural curvature profilebased at least on the resulting flexural stiffness profile and the totalload profile; and modifying the camber profile based at least on theresulting flexural curvature profile and the desired total curvatureprofile.

In some embodiments, calculating the torsional deformation profileinvolves: determining a resulting torsional stiffness profile of thegliding board based at least on the at least one stiffness designvariable profile, the width profile, and at least one gliding boardmaterial property; and generating the torsional deformation profilebased at least on the resulting torsional stiffness profile and thetotal load profile.

In some embodiments, the total load profile includes a snow loadprofile, and wherein generating the total load profile involvescalculating the snow load profile by: determining a desired snowpenetration depth profile such that a total magnitude of the resultingsnow load profile is approximately equal to the total magnitude of arelevant component of the athlete load profile, and such that a positionof a centroid of the resulting snow load profile is approximately equalto a position of a centroid of the relevant component of the athleteload profile.

In some embodiments, the total load profile includes the athlete loadprofile.

In some embodiments, the total load profile includes at least onebinding load profile.

In some embodiments, modifying the desired total curvature profileinvolves: determining a resulting deformed shape of the gliding boardduring the carved turn based at least on the desired total curvatureprofile and the torsional deformation profile; determining a positionand an orientation for the resulting deformed shape of the glidingboard, wherein the position and orientation correspond to the desiredsnow penetration depth profile; determining a resulting snow traceprofile based at least on the resulting deformed shape of the glidingboard and the determined position and orientation; determining aresulting global curvature profile based at least on the resulting snowtrace profile; comparing the resulting global curvature profile to thedesired global curvature profile; and modifying the desired totalcurvature profile based at least on the comparison between the resultingglobal curvature profile and the desired global curvature profile.

In some embodiments, determining the position and the orientation forthe resulting deformed shape of the gliding board involves: determiningat least one anchor point along a length of the gliding board; anddetermining the position and the orientation for the resulting deformedshape of the gliding board such that, at each of the at least one anchorpoint, a resulting snow penetration depth is approximately equal to acorresponding desired snow penetration depth value from the desired snowpenetration depth profile.

In some embodiments, modifying the sidecut profile involves: determininga resulting deformed shape of the gliding board during the carved turnbased at least on the desired total curvature profile and the torsionaldeformation profile; determining a position and an orientation for theresulting deformed shape of the gliding board, wherein the position andorientation correspond to the desired snow penetration depth profile;and modifying the sidecut profile such that a resulting snow penetrationdepth profile is approximately equal to the desired snow penetrationdepth profile.

In some embodiments, determining the position and the orientation forthe resulting deformed shape of the gliding board involves: determiningat least one anchor point along a length of the gliding board; anddetermining the position and the orientation for the resulting deformedshape of the gliding board such that, at each of the at least one anchorpoint, a resulting snow penetration depth is approximately equal to acorresponding desired snow penetration depth value from the desired snowpenetration depth profile.

In some embodiments modifying at least the width profile, the sidecutprofile and at least one of the at least one stiffness design variableprofile at least once further involves: modifying the total load profilebased at least on the athlete load profile.

In accordance with some embodiments, there is provided a system forgenerating a design for a gliding board. The system includes a memoryand a processor. The memory can store a width profile, a camber profile,and at least one stiffness design variable profile. The processor isoperable to: define a desired carved turn of the gliding board, thedesired carved turn being defined at least by a nominal edging angle andan athlete load profile, wherein the athlete load profile represents aload that is applied by an athlete to the gliding board during thedesired carved turn; define a desired global curvature profile, whereinthe desired global curvature profile corresponds to a desired snow traceprofile for the desired carved turn; generate a desired deformed shapeof the gliding board during the desired carved turn, the desireddeformed shape of the gliding board being defined at least by a desiredtotal curvature profile, wherein the desired total curvature profile isinitially set to correspond to the desired global curvature profile;generate a sidecut profile of the gliding board; generate a widthprofile of the gliding board based at least on the sidecut profile;generate a camber profile of the gliding board; generate at least onestiffness design variable profile, wherein the at least one stiffnessdesign variable profile, in conjunction with at least the width profileand at least one gliding board material property, dictates a resultingflexural stiffness profile and a resulting torsional stiffness profileof the gliding board; generate a total load profile based at least onthe athlete load profile, wherein the total load profile represents atotal load that is applied to the gliding board during the desiredcarved turn, and wherein generating the total load profile involvesgenerating a desired snow penetration depth profile; modify at least thewidth profile, the sidecut profile and at least one of the at least onestiffness design variable profile at least once; and define the designfor the gliding board based at least on the width profile, the camberprofile, and the at least one stiffness design variable profile. Theprocessor can modify at least the width profile, the sidecut profile andat least one of the at least one stiffness design variable profile atleast once by: calculating a desired flexural stiffness profile of thegliding board based at least on the total load profile, such that thedesired flexural stiffness profile approximately achieves the desiredtotal curvature profile during the carved turn; modifying at least oneof the at least one stiffness design variable profile such that theresulting flexural stiffness profile is approximately equal to thedesired flexural stiffness profile; calculating a torsional deformationprofile of the gliding board during the carved turn based at least onthe total load profile; modifying the desired total curvature profilebased at least on the torsional deformation profile in order to achievea resulting global curvature profile that is approximately equal to thedesired global curvature profile during the carved turn; modifying thesidecut profile based at least on the torsional deformation profile inorder to achieve a resulting snow penetration depth profile that isapproximately equal to the desired snow penetration depth profile; andmodifying the width profile based at least on the modified sidecutprofile.

In some embodiments, modifying at least the width profile, the sidecutprofile and at least one of the at least one stiffness design variableprofile at least once involves: iteratively repeating the steps ofcalculating the desired flexural stiffness profile, modifying at leastone of the at least one stiffness design variable profile, calculatingthe torsional deformation profile, modifying the desired total curvatureprofile, modifying the sidecut profile, and modifying the width profileuntil a difference between at least one of the width profile, thesidecut profile, and at least one of the at least one stiffness designvariable profile associated with a current iteration and at least one ofthe width profile, the sidecut profile, and at least one of the at leastone stiffness design variable profile associated with the previousiteration is less than a predetermined threshold.

In some embodiments, defining the desired global curvature profileinvolves: defining the desired global curvature profile to correspond toa carved turn that exhibits an approximately constant turning radius.

In some embodiments, each of the desired global curvature profile andthe resulting global curvature profile embodies a curvilinear form thatcan be determined by projecting a curvilinear geometry of acorresponding snow trace profile onto an oblique projection plane thatis inclined with respect to the plane on which the respective snow traceprofile exists.

In some embodiments, generating the sidecut profile involves: defining adesired geometry of a snow penetration depth profile, wherein the snowpenetration depth profile represents the penetration depth of thegliding board into snow during the desired carved turn; and generatingthe sidecut profile based at least on the desired global curvatureprofile, the desired geometry of the snow penetration depth profile, andthe nominal edging angle.

In some embodiments, defining the desired geometry of the snowpenetration depth profile involves: defining the desired geometry of thesnow penetration depth profile to approximately follow a linear functionover a length of an effective base region of the gliding board.

In some embodiments, generating the camber profile involves: determininga first assumed total curvature profile of the gliding board and a firstload profile of the gliding board corresponding to a first gliding boardedging angle; determining a second assumed total curvature profile ofthe gliding board and a second load profile of the gliding boardcorresponding to a second gliding board edging angle; and generating thecamber profile such that it facilitates a gliding board design that iscapable of satisfying the first assumed total curvature profile with thefirst load profile, and is also capable of satisfying the second assumedtotal curvature profile with the second load profile.

In some embodiments, at least one of the at least one stiffness designvariable profile includes a core thickness profile of the gliding board.

In some embodiments, calculating the desired flexural stiffness profileinvolves: calculating the desired flexural stiffness profile of thegliding board based at least on the camber profile, the desired totalcurvature profile, and the total load profile.

In some embodiments, modifying at least one of the at least onestiffness design variable profile involves: finding the at least onestiffness design variable profile such that the resulting flexuralstiffness profile is approximately equal to the desired flexuralstiffness profile, wherein the at least one stiffness design variableprofile includes a plurality of local stiffness design variable values,wherein each local stiffness design variable value is limited to apredetermined range of allowable values.

In some embodiments, modifying at least the width profile, the sidecutprofile and at least one of the at least one stiffness design variableprofile at least once further involves modifying the camber profile atleast once by: determining a resulting flexural stiffness profile of thegliding board based at least on the at least one stiffness designvariable profile, the width profile, and at least one gliding boardmaterial property; determining a resulting flexural curvature profilebased at least on the resulting flexural stiffness profile and the totalload profile; and modifying the camber profile based at least on theresulting flexural curvature profile and the desired total curvatureprofile.

In some embodiments, calculating the torsional deformation profileinvolves: determining a resulting torsional stiffness profile of thegliding board based at least on the at least one stiffness designvariable profile, the width profile, and at least one gliding boardmaterial property; and generating the torsional deformation profilebased at least on the resulting torsional stiffness profile and thetotal load profile.

In some embodiments, the total load profile includes a snow loadprofile, and wherein generating the total load profile involvescalculating the snow load profile by: determining a desired snowpenetration depth profile such that a total magnitude of the resultingsnow load profile is approximately equal to the total magnitude of arelevant component of the athlete load profile, and such that a positionof a centroid of the resulting snow load profile is approximately equalto a position of a centroid of the relevant component of the athleteload profile.

In some embodiments, the total load profile includes the athlete loadprofile.

In some embodiments, the total load profile includes at least onebinding load profile.

In some embodiments, modifying the desired total curvature profileinvolves: determining a resulting deformed shape of the gliding boardduring the carved turn based at least on the desired total curvatureprofile and the torsional deformation profile; determining a positionand an orientation for the resulting deformed shape of the glidingboard, wherein the position and orientation correspond to the desiredsnow penetration depth profile; determining a resulting snow traceprofile based at least on the resulting deformed shape of the glidingboard and the determined position and orientation; determining aresulting global curvature profile based at least on the resulting snowtrace profile; comparing the resulting global curvature profile to thedesired global curvature profile; and modifying the desired totalcurvature profile based at least on the comparison between the resultingglobal curvature profile and the desired global curvature profile.

In some embodiments, determining the position and the orientation forthe resulting deformed shape of the gliding board involves: determiningat least one anchor point along a length of the gliding board; anddetermining the position and the orientation for the resulting deformedshape of the gliding board such that, at each of the at least one anchorpoint, a resulting snow penetration depth is approximately equal to acorresponding desired snow penetration depth value from the desired snowpenetration depth profile.

In some embodiments, modifying the sidecut profile involves: determininga resulting deformed shape of the gliding board during the carved turnbased at least on the desired total curvature profile and the torsionaldeformation profile; determining a position and an orientation for theresulting deformed shape of the gliding board, wherein the position andorientation correspond to the desired snow penetration depth profile;and modifying the sidecut profile such that a resulting snow penetrationdepth profile is approximately equal to the desired snow penetrationdepth profile.

In some embodiments, determining the position and the orientation forthe resulting deformed shape of the gliding board involves: determiningat least one anchor point along a length of the gliding board; anddetermining the position and the orientation for the resulting deformedshape of the gliding board such that, at each of the at least one anchorpoint, a resulting snow penetration depth is approximately equal to acorresponding desired snow penetration depth value from the desired snowpenetration depth profile.

In some embodiments modifying at least the width profile, the sidecutprofile and at least one of the at least one stiffness design variableprofile at least once further involves: modifying the total load profilebased at least on the athlete load profile.

BRIEF DESCRIPTION OF THE DRAWINGS

Several embodiments will now be described in detail with reference tothe drawings, in which:

FIG. 1 shows a flowchart of a method for operating a ski design system,in accordance with an example embodiment;

FIG. 2 shows a block diagram of a ski design system, in accordance withan example embodiment;

FIG. 3A shows an elevation view of an example ski, in accordance with anexample embodiment;

FIG. 3B shows a planform view of the ski shown in FIG. 3A, in accordancewith an example embodiment;

FIG. 4 shows an elevation view of an example boot-binding system for theski shown in FIG. 3A, in accordance with an example embodiment;

FIG. 5A shows an elevation view of an example boot for the boot-bindingsystem shown in FIG. 4, in accordance with an example embodiment;

FIG. 5B shows an elevation view of various example components of anexample binding for the boot-binding system shown in FIG. 4, inaccordance with an example embodiment;

FIG. 6A shows an elevation view of a portion of the boot shown in FIG.5A, illustrating a rear forward pressure force, in accordance with anexample embodiment;

FIG. 6B shows an elevation view of a portion of the binding system shownin FIG. 5B, illustrating a rear forward pressure force, in accordancewith an example embodiment;

FIG. 7A shows an elevation view of a portion of the boot shown in FIG.5A, illustrating a heel-cup clamping force, a rear AFD (anti-frictiondevice) perch force, and a brake pedal force, in accordance with anexample embodiment;

FIG. 7B shows an elevation view of a portion of the binding system shownin FIG. 5B, illustrating a heel-cup clamping force, a rear AFD perchforce, and a brake pedal force, in accordance with an exampleembodiment;

FIG. 8A shows an elevation view of a portion of the boot shown in FIG.5A, illustrating a front forward pressure force, in accordance with anexample embodiment;

FIG. 8B shows an elevation view of a portion of the binding system shownin FIG. 5B, illustrating a front forward pressure force, in accordancewith an example embodiment;

FIG. 9A shows an elevation view of a portion of the boot shown in FIG.5A, illustrating a toe-cup clamping force and a front AFD perch force,in accordance with an example embodiment;

FIG. 9B shows an elevation view of a portion of the binding system shownin FIG. 5B, illustrating a toe-cup clamping force and a front AFD perchforce, in accordance with an example embodiment;

FIG. 10 shows a cross-sectional view of the ski shown in FIG. 3,illustrating a sandwich-type construction, in accordance with an exampleembodiment;

FIG. 11 shows a graph illustrating an example relationship betweenflexural stiffness, core thickness, and width, in accordance with anexample embodiment;

FIG. 12 shows a flowchart of a method for operating a ski design system,in accordance with an example embodiment;

FIG. 13 shows a flowchart of a method for operating a ski design system,in accordance with an example embodiment;

FIG. 14 shows a flowchart of a method for operating a ski design system,in accordance with an example embodiment;

FIG. 15 shows a graph illustrating a relationship between normal forcesapplied to a ski and the absolute value of the tangent of thecorresponding edging angles of the ski;

FIG. 16 shows a flowchart of a method for operating a ski design system,in accordance with an example embodiment; and

FIG. 17 shows a flowchart of a method for operating a ski design system,in accordance with an example embodiment.

The drawings, described below, are provided for purposes ofillustration, and not of limitation, of the aspects and features ofvarious examples of embodiments described herein. For simplicity andclarity of illustration, elements shown in the drawings have notnecessarily been drawn to scale. The dimensions of some of the elementsmay be exaggerated relative to other elements for clarity. It will beappreciated that for simplicity and clarity of illustration, whereconsidered appropriate, reference numerals may be repeated among thedrawings to indicate corresponding or analogous elements or steps.

DESCRIPTION OF EXAMPLE EMBODIMENTS

The various embodiments described herein generally relate to systems andmethods for generating a design for a ski. The ski design systemdisclosed herein can generate ski designs which are customized oroptimized for a particular athlete and/or for particular conditions. Theski design system can generate ski designs which can achieve aparticular snow trace geometry and a particular snow pressuredistribution.

In order to prescribe a flexural stiffness profile that varies over thelength of the ski, as well as a torsional stiffness profile that variesover the length of the ski, it may be necessary to define at least one“stiffness design variable”; at any position along the length of theski, the flexural stiffness and torsional stiffness of the ski can thenbe dictated by the relationship between the local value of thisstiffness design variable, the local width of the ski, the constructionarchitecture of the ski, and the mechanical properties of the materialsthat are present within the ski construction. The “stiffness designvariable profile” defines the value of the relevant stiffness designvariable at any position along the length of the ski.

For example, skis that employ sandwich-type construction (which isdescribed below) commonly employ core thickness as the primary stiffnessdesign variable. In other words, the thickness of the core of the skivaries over the length of the ski in order to achieve a desired flexuralstiffness value and torsional stiffness value at each position along thelength of the ski. In such embodiments, at least one of the at least onestiffness design variable profile is represented by a core thicknessprofile.

Most of the example embodiments discussed herein employ core thicknessas the only stiffness design variable; however, it should be understoodthat a broad range of alternative stiffness design variables could beemployed by alternative embodiments. For example, alternativeembodiments could employ a ski construction that is of substantiallyconstant thickness over its entire length, but features longitudinallyoriented stiffening ribs on its upper surface; in this case, thestiffness design variable could take the form of one or morecharacteristic of these stiffening ribs (such as the height and/or widthof each rib), such that the targeted flexural and/or torsional stiffnessat any position along the length of the ski can be achieved by adjustingsaid variable characteristic(s) of the stiffening ribs. It should alsobe understood that a ski could be designed such that it exhibits two ormore zones over its length, wherein two or more of these zones differ intheir construction architecture and/or their dominant stiffness designvariable.

In order to design a ski that is capable of simultaneously achieving anideal snow trace geometry and an ideal snow pressure distribution, it isdesirable to optimize the design of the ski such that a harmoniousrelationship exists between its sidecut profile, its camber profile, itsflexural stiffness profile, and its torsional stiffness profile. Theaforementioned optimization is further complicated by the fact that theflexural stiffness and torsional stiffness profiles of the ski can becoupled to each other in a manner that is dependent upon therelationship between the at least one stiffness design variable profile,the width profile, the construction architecture of the ski, and themechanical properties of the materials that are present within the skiconstruction. Furthermore, practical limitations may prohibit the use ofan ideal flexural stiffness profile—for example, it is not practicableto build a ski that exhibits a thickness profile that tapers to zero.The example embodiments described herein provide a ski design system forthe design of alpine skis in a manner that accurately captures the manystructural, physical, and geometric phenomenon that govern alpine skicarving mechanics.

The ski design system disclosed herein can use iterative or recursivemethods to simultaneously consider the relationship between the widthprofile, the camber profile, and the at least one stiffness designvariable profile of the ski, while at the same time avoiding divergence.As will be described below, the ski design system can consider thepractical limitations of ski construction and compensate the ski designaccordingly. The ski design system can generate a design for a ski thatmaximizes performance for a particular athlete, under particularconditions, by minimizing drag while maximizing grip during carvedturns.

The term “optimization” is used herein to describe a process ormethodology of improving an aspect of a ski design to a desired extent,while accounting for relevant conditions and constraints. The presentinventor recognizes that recent literature frequently uses the term“optimization” in reference to mathematical optimization, whereinmathematical procedures (numerical and/or analytical) are employed tofind the global or local maxima or minima of a mathematical function.Conversely, the present descriptions herein employ the more generalmeaning of “optimization” as any process or methodology (mathematical orotherwise) of achieving a notably improved result. This distinction isimportant, as some embodiments can achieve a reasonably optimized skidesign without employing the types of mathematical optimizationtechniques that are commonly cited in general literature that makes useof the term “optimization”.

The term “gliding board” is used herein in reference to an apparatusthat is intended to slide along low-friction surfaces (such as snow orice-covered terrain). A gliding board may feature one or more bootretention systems (bindings) to which an athlete can fasten a boot(shoe) or boots (shoes), thus facilitating a connection between thegliding board and one or both of the athlete's feet. Some gliding boardsmay utilize an intermediary system or structure between the athlete andthe gliding board; for example, so called “ski-bikes” and “ski-scooters”feature structural frames that are mounted to the upper surface of thegliding board(s), and the athlete interacts primarily with saidstructural frames. Typical examples of such gliding boards include (butare not limited to): alpine (downhill) skis, snowboards, skiboards,mono-skis, toboggans, sit-skis, ski-bikes, ski-scooters, Nordic skis(including, but not limited to, cross-country skis, telemark skis, androller skis), and other similar equipment. Some gliding board activities(such as alpine skiing) require the use of more than one gliding board;as such, the term “gliding board” may refer to a single stand-alonegliding board (such as a snowboard), but it also may refer to one of apair of gliding boards (such as a pair of alpine skis), or one of alarger number of corresponding gliding boards that are intended to beused together. While the following description is presented in thecontext of alpine (downhill) skis, some embodiments may includegenerating a design for other types of gliding boards. In particular,some embodiments may facilitate the design of athlete-customizedsnowboards. Accordingly, it should be appreciated that the term “ski” isused herein for ease of exposition, and may, in some embodiments, besubstituted by the term “gliding board”.

The present document describes a variety of quantitative variables(physical, mechanical, and geometric) that vary along the length of theski; the term “profile” is used herein in reference to the collection ofdata that describes how the value of each such quantitative variable canbe found at any position along the length of the ski. Stated anotherway, the term “profile” is used herein to denote the topography of acertain quantitative variable (physical, mechanical, or geometric) alongthe length of the ski. For example, the “width profile” of the skidefines the width of the ski at any position along its length.Similarly, the “flexural curvature profile” of the ski defines theflexural curvature exhibited by the ski at any position along itslength.

The present descriptions herein use the term “elevation” in reference toany projection plane (viewing plane) that is parallel to thelongitudinal axis of the ski and perpendicular to the base surface ofthe ski. As such, any geometry that is described in the context of beingviewed in elevation shall be assumed to have been projected onto a planethat is parallel to the longitudinal axis of the ski and perpendicularto the base surface of the ski. In such an elevation projection plane,the “upward” direction denotes the direction that is approximatelyperpendicular to the base surface of the ski, and points away from thesnow. The term “elevation profile” is used herein in reference to sometwo-dimensional geometric contours that exist on a plane that isparallel to the longitudinal axis of the ski and perpendicular to thebase surface of the ski.

The present descriptions herein make reference to numerous angles, andsome of said angles are referenced in equations. It should be understoodthat all of said angles referenced in equations are intended to beexpressed in units of radians. All equations, formulae, and mathematicalexpressions incorporated herein were formulated under the assumptionthat said angles are expressed in units of radians.

Referring now to FIGS. 3A and 3B, shown therein is an elevation view anda planform view, respectively, of an example alpine ski 3000. Asdepicted in FIGS. 3A and 3B, alpine skis can generally be characterizedas long, narrow, and thin leaf spring elements having a tip 3300(forward extremity), a tail 3500 (rearward extremity), and aboot-binding system (not shown) for affixing the athlete's foot to thetop surface of the ski 3000 at a position that is typically slightly aftof the mid-point between the tip 3300 and tail 3500 of the ski 3000. Thetip 3300 of the ski features a shovel region 3100, which ischaracterized as having an upward curve (concave upward geometry) whenviewed in elevation. Similarly, the tail 3500 of the ski often alsofeatures an upward curve (concave upward geometry) when viewed inelevation; however, the upward curve of the tail 3500 is often lesspronounced than that present at the shovel 3100. The base 3610 is thebottom surface of the ski, which makes direct contact with the snow (seeFIG. 10 for additional detail).

The main running length of the ski that interacts most intimately withthe snow is the region that lies between the aforementioned upwardcurves at the shovel 3100 and tail 3500 of the ski. In the case ofalpine skis that are intended for racing and/or on-piste carving, thismain running length region often exhibits a cambered geometry, whereinthe base 3610 of the ski exhibits a concave downward (convex upward)geometry when viewed in elevation. In some cases, some regions of themain running length of the ski may exhibit rocker (reverse-camber),wherein the base 3610 of the ski exhibits a concave upward (convexdownward) geometry when viewed in elevation. Most alpine skis that areintended for racing and/or on-piste carving exhibit predominantlycambered geometries within their main running lengths; however, it isbecoming increasingly common for such skis to feature small regions ofrockered geometry.

The “camber elevation profile” denotes the overall geometry of the basesurface of the unloaded ski, when viewed in elevation, wherein saidelevation profile exists on a plane that is parallel to the longitudinalaxis of the ski and perpendicular to the base surface of the ski. The“camber curvature profile” of the ski defines the curvature of thecamber elevation profile at any position along the length of the ski,wherein said curvature is measured within a plane that is parallel tothe longitudinal axis of the ski and perpendicular to the base surfaceof the ski. The camber elevation profile may include some combination ofcambered regions and rockered regions in order to achieve the desiredcharacteristics of the ski.

The edging angle of a ski is defined as the angle between the base ofthe ski and the surface upon which the ski is gliding (for example: thesurface of the snow), measured about an axis that is approximatelyparallel to the longitudinal axis of the ski.

The inclination angle of an athlete is defined as the angle between thesurface upon which the ski is gliding (for example: the surface of thesnow) and a straight line drawn between the centre-of-mass of theathlete and the centroid of the total snow pressure distribution thatacts upon both skis, measured about an axis that is parallel to theinstantaneous velocity vector of the athlete (which is approximatelyparallel to the longitudinal axis of the skis). As such, it can bereasoned that the inclination angle is equal to the angle between thesurface upon which the ski is gliding (for example: the surface of thesnow) and the total force vector that the athlete is imposing upon thesnow, measured about an axis that is parallel to the instantaneousvelocity vector of the athlete (which is approximately parallel to thelongitudinal axis of the skis).

The angulation angle of the athlete is then defined as the differencebetween the edging angle and the inclination angle.

When a ski is pressed downward onto a flat surface by a force that isapplied in the vicinity of the boot-binding system, wherein said forceacts approximately perpendicular to the base surface of the ski, the skiwill typically become bent in a direction that opposes the downwardconcavity of its camber elevation profile such that at least one of itsedges makes contact with said flat surface over a majority of the ski'slength (excluding the shovel and tail regions that exhibit upwardcurves); when this occurs, the ski is said to be “de-cambered”. Thedegree to which a ski is de-cambered onto a flat surface depends uponthe edging angle of the ski on said flat surface. A ski that isde-cambered onto a flat surface with an edging angle of zero willtypically adopt a quasi-flat base surface over a majority of the ski'slength (excluding the shovel and tail regions that exhibit upwardcurves), since the ski will deform until its base comes into contactwith the flat surface. Conversely, a ski that is de-cambered onto a flatsurface with an edging angle that is non-zero will generally adopt acurved base geometry, wherein the base of the ski exhibits a concaveupward (convex downward) geometry when viewed in elevation.

Continuing with reference to FIGS. 3A and 3B, the effective base 3200 ofthe ski 3000 is often defined as the region of the ski's base 3610 thatwould contact a flat rigid surface if the ski 3000 were to be fullyde-cambered onto said surface with an edging angle of zero. As such, theeffective base region 3200 typically excludes the shovel 3100 and tailregions 3500 that exhibit upward curves. The length of the effectivebase region is typically referred to as the “effective base length”. Inthe following descriptions herein, the term “mid-base” denotes themid-point between the forward-most point within the effective baseregion of the ski and the rearward-most point within the effective baseregion of the ski. The term “forebody” is used herein in reference tothe region of the ski that is located forward of the mid-base point, andthe term “afterbody” is used herein in reference to the region of theski that is located aft of the mid-base point.

As depicted in FIGS. 3A and 3B, an alpine ski that is intended forracing and on-piste carving generally exhibits a waisted (hour-glassshaped) planform geometry, wherein said ski is relatively narrow nearits mid-base and is wider in the vicinity of its tip 3300 and tail 3500.The waist 3400 of the ski 3000 is defined as the narrowest point of theski within its effective base region 3200, and is typically positionedslightly aft of the mid-base of the ski 3000.

This waisted planform geometry is generally achieved through thecreation of curved sidecuts along either side of the ski. The sidecut oneach edge of the ski characterizes the extent to which the planformgeometry of the corresponding edge of the ski deviates from a straightline. At each position along the length of the ski, the local “sidecutdepth” represents the distance between the local position of the edge ofthe ski and a straight line drawn from the tip of the ski to the tail ofthe ski, wherein said straight line is parallel to the longitudinal axisof the ski and is tangent to the relevant edge of the ski at the widestpoint of the ski, and wherein the local sidecut depth is measuredperpendicular to said straight line.

The “sidecut profile” of the ski defines the local sidecut depth of theski at any position along the length of the ski; as such, the sidecutprofile represents the planform geometry of one of the edges of the ski.While the sidecut profile of a ski is not necessarily of constant radiusgeometry, the nominal radius of curvature of the sidecut geometry(denoted herein by “nominal sidecut radius”) is often reported as theradius of an arc that can be fitted through three points along one ofthe edges of the ski 3000, wherein one of said points is positioned ator near the waist 3400, one of said points is positioned slightly aft ofthe widest point of the ski 3000 that is forward of the waist 3400, andone of said points is positioned slightly forward of the widest point ofthe ski 3000 that is aft of the waist 3400.

The taper of the ski 3000 is defined as the difference between the widthof the widest point of the ski 3000 that is forward of the waist 3400and the width of the widest point of the ski that is aft of the waist3400.

The “width profile” of the ski defines the width of the ski at anyposition along the length of the ski. If the right hand and left handsidecut profiles of the ski are symmetrical about the longitudinal axisof the ski, then the sidecut profile of the ski can be calculated as afunction of the width profile of the ski. For ease of exposition, mostof the example embodiments discussed herein will assume that the righthand and left hand sidecut profiles of the ski are symmetrical about thelongitudinal axis of the ski; however, this should not be consideredrestrictive, as some embodiments could also be applied to the design ofskis that do not have right hand and left hand sidecut profiles of theski that are symmetrical about the longitudinal axis of the ski.

A broad variety of boot-binding systems exist for the various types ofgliding board sports. Within the context of downhill skiing, therecurrently exist a few different types of binding systems, including:alpine bindings, alpine touring bindings, and telemark bindings.Although each type of binding is ultimately intended to serve the samegeneral purpose of fastening a boot to the upper surface of a ski, eachtype of binding does so using a unique mechanical architecture which, inturn, imposes a unique set of loads upon the ski. While some embodimentscould be used with any boot-binding architecture, the followingdescriptions herein will focus on the use of alpine ski bindings thatcomply with the International Organization for Standardization (ISO)specification ISO 9462, as well as the use of alpine ski boots thatcomply with ISO specification ISO 5355.

Referring now to FIGS. 4-9, shown therein are elevation views of anexample boot-binding system 4000. As depicted in FIGS. 4, 5A, and 5B theboot-binding system 4000 is based upon the use of a boot 4100 thatfeatures a protrusion referred to as a toe lug 4120 at the toe of theboot 4100, and another protrusion referred to as a heel lug 4160 at theheel of the boot 4100. The boot 4100 rests upon a pair of AFD(anti-friction device) perches 4230, 4250 positioned near the toe andheel of the boot.

As depicted in FIGS. 7A and 7B, the heel lug 4160 of the boot is heldfirmly to the ski by a heel-cup 4260, which applies a clamping force4267 (F_(Cup_r)) to the heel-cup 4260. This clamping force is typicallycreated by a spring-actuated articulation of the heel-cup about aheel-cup pivot axis 4270.

As depicted in FIGS. 9A and 9B, the toe lug 4120 of the boot 4100 isheld firmly to the ski by a toe-cup 4220, which in many cases applies aclamping force 4227 (F_(Cup_f)) to the toe-cup 4220 (some alpine bindingsystems do not apply a clamping force at the toe-cup). This clampingforce is typically created by a spring-actuated articulation of thetoe-cup 4220 about a toe-cup pivot axis 4210; however, some bindingsystems employ different mechanisms to achieve the toe-cup clampingforce.

As depicted in FIGS. 6A, 6B, 8A and 8B, the boot 4100 is compressedlongitudinally (parallel to the sole 4140 of the boot) between thetoe-cup 4220 and the heel-cup 4260; this longitudinal compression forceis referred to as the forward pressure loading. The forward pressureloading is generated by a forward pressure force 4265 that is applied atthe heel-cup (F_(FP_r)), and is equilibrated by another forward pressureforce 4225 at the toe-cup (F_(FP_f)).

As depicted in FIGS. 5A and 5B, the heel-cup 4260 of the binding isconnected to a heel carriage 4280 that slides within a heel track 4290,wherein said heel track is substantially parallel to the longitudinalaxis of the ski 3000. The heel carriage 4280 is acted on by anadjustable forward pressure spring, which is responsible for generatingthe necessary forward pressure force. In most cases, the forwardpressure spring is linked to a point that is positioned in the vicinityof the heel of the boot; however, in some binding systems, the forwardpressure spring is connected to a tether (toe-heel link) that isfastened to a point on the ski that is near the mounting point of thetoe-cup 4220, which renders the forward pressure system somewhat lesssensitive to flexural deformations of the ski.

Skis that comply with the aforementioned ISO specifications feature abraking device that will prevent the ski from sliding down the hill inthe event that the boot is released from its binding. Continuing withreference to FIGS. 5A and 5B, these brake systems are typically actuatedby spring-loaded pedals 4240 that are positioned beneath the boot,slightly forward of the rear AFD perch 4250. As depicted in FIGS. 7A and7B, due to the spring-loaded nature of this brake pedal 4240, thereexists a brake pedal force 4245 (F_(BP)) between the pedal 4240 and thebottom of the sole 4140 of the boot 4100.

Finally, as depicted in to FIGS. 7A, 7B, 9A and 9B, the front AFD perch4230 and the rear AFD perch 4250 each apply reaction forces 4235 and4255 (F_(AFB_f) and F_(AFD_r), respectively) in order to equilibrate theaforementioned binding loadings, as well as any other unbalanced snowloadings, gravity loadings, and/or inertial loadings that are imposedupon the ski 3000. In general, the aforementioned binding loads aretypically generated by passive mechanical spring systems; as such, for acontemporary alpine binding system, it is possible to perform a seriesof mechanical tests that will provide load-versus-deflection responsesof each component of the binding system, thus facilitating thecalculation of each of the binding loads as a function of the displacedpositions of each binding component.

A broad variety of gliding board construction architectures exist forthe various types of gliding board sports. Within the context ofdownhill skiing, the most commonly used construction architecturesinclude: sandwich-type (sidewall), cap, and semi-cap (half-cap).Sandwich-type construction is generally most conducive for the creationof skis that exhibit high torsional stiffness; as such, sandwich-typeconstruction is often used for race skis and high-performance on-pistecarving skis. Although the present description describes ski designshaving a sandwich-type construction, it should be noted that someembodiments may include generating designs that include other glidingboard construction architectures.

Referring now to FIG. 10, shown therein is a cross-sectional view of aski that exhibits sandwich-type construction 3600; the cross-sectionalview of the ski is drawn on a plane that is oriented perpendicular tothe longitudinal axis of the ski, and therefore depicts the width andthickness dimensions of the ski's cross-sectional geometry. As depictedin FIG. 10, the base 3610 is the bottom surface of the ski that comesinto direct contact with the snow; the base 3610 is typically formed ofan ultra-high molecular weight polyethylene (UHMW-PE) material, whichoften contains additives to reduce friction. The edges 3620 flank theboth sides of the bottom surface 3610 of the ski; the edges 3620 aretypically composed of a hardened steel material. The core 3630 isenveloped within all of the other constituents of the ski construction.The core 3630 is most commonly formed of a laminate of one or more typesof wood; however, some skis employ other core materials, such as:polymeric foams, honeycomb materials, and hybrid laminates of a fewdifferent types of materials. The lower reinforcing plies 3640 arepositioned directly below the core 3630, whereas the upper reinforcingplies 3650 are positioned directly above the core 3630. These lower andupper reinforcing plies 3640, 3650 typically include fibre-reinforcedpolymer layers (such as fibreglass, carbon-fibre, and/or aramid fibrereinforced thermoset resins), and often also include layers of sheetmetal (aluminium and/or steel alloys) as well. The topsheet 3660 is anon-structural layer that is responsible for protecting the upperreinforcing plies, and providing aesthetic adornment to the uppersurface of the ski. The topsheet 3660 is typically formed of athermoplastic material, such as a polyamide or a UHMW-PE material. Thesidewalls 3670 are responsible for sealing the sides of the core 3630,which would otherwise be exposed between the lower and upper reinforcingplies 3640, 3650.

The assembly of the lower reinforcing plies 3640, the upper reinforcingplies 3650, and the sidewalls 3670 constitutes a closed box-section(closed thin-walled tube), which may offer considerable structuralbenefits; in the context of torsional loading, this type of structuralmember functions as a torsion-box, wherein each of the skins of the boxsection develops shear flow along its mid-plane, thus enabling thetorsion-box to resist torsional deformations in a very efficient manner.The sidewalls 3670 are typically formed of either a thermoplasticmaterial (such as acrylonitrile butadiene styrene or UHMW-PE) or a paperreinforced thermoset resin material (such as kraft paper reinforcedphenol formaldehyde resin). Due to the relatively high shear modulus andlow toughness of paper reinforced thermoset resins, the use of thesematerials as sidewalls typically yields highly efficient torsion-boxbehaviour, but results in relatively poor durability; as such, this typeof sidewall material is typically only used in race skis and somehigh-performance on-piste carving skis.

FIG. 10 further shows the overall width of the ski 3680 (measuredbetween the outer surfaces of the edges 3620), as well as the thicknessof the core of the ski 3690.

The sidecut profile of the ski is dictated by variations in the width ofthe ski 3680 (the width profile) over the length of the ski. As thewidth of the ski 3680 varies, the widths of the base 3610, lowerreinforcing plies 3640, core 3630, upper reinforcing plies 3650, andtopsheet 3660 also vary. Conversely, the widths of the sidewalls 3670and edges 3620 typically remain constant over the length of the ski.

In order to achieve a targeted flexural stiffness profile (bendingstiffness profile) over the length of the ski, the thickness of the skimay be varied over its length. This variation in thickness is typicallyachieved by varying the thickness of the core 3690, and consequently,the height of each sidewall 3670 is also varied. As such, it should beunderstood that core thickness can be selected as the stiffness designvariable for this particular example embodiment. Conversely, the base3610, lower reinforcing plies 3640, upper reinforcing plies 3650, andtopsheet 3660 typically maintain a constant thickness over the length ofthe ski. In some cases, some or all of the lower and/or upperreinforcing plies 3640, 3650 may exhibit varying thicknesses over thelength of the ski as well, but this is relatively uncommon.

The “thickness profile” of the ski defines the total thickness of theski at any position along the length of the ski; similarly, the “corethickness profile” defines the thickness of the core of the ski at anyposition along the length of the ski. Core thickness has been selectedas the only stiffness design variable for most of the exampleembodiments that are discussed below; as such, the core thicknessprofile represents the only stiffness design variable profile that isemployed by most of the example embodiments that are discussed below.However, it should be noted that other stiffness design variables arepossible.

A curvilinear tangential coordinate system (XmidB) is defined along thelongitudinal axis of the base of the ski (centred on the width of theski), and follows the base contours of the ski as the ski changes shapefrom its cambered state to its deformed shape under loaded conditions.The origin of the tangential coordinate system (XmidB) is positioned atthe mid-base point (mid-point between the forward and aft extremities ofthe effective base region of the ski). A global three-dimensionalCartesian coordinate system is then defined, wherein the X-axis isparallel to the tangential coordinate system (XmidB) at the mid-basepoint of the ski and points rearward, the Y-axis is perpendicular to theX-axis and parallel to the surface of the snow, the Z-axis isperpendicular to the X-Y plane and points in the upward direction (awayfrom the snow), and the origin is positioned at the mid-base point ofthe ski. At each longitudinal position along the length of the ski, theorientation of the base surface of the ski is represented by three localorthogonal direction vectors (orientation vectors) of unit length: the vvector is aligned with the longitudinal tangential axis of the ski(parallel to the local base contour of the ski), and points in therearward direction; the n vector is perpendicular to the base surface ofthe ski, and points in the direction that is opposite the snow(generally upwards); and the b vector is perpendicular to both the v andn vectors.

The present descriptions herein often represent curvilinear geometriesin terms of their curvature profiles, wherein a curvature profiledefines the curvature of the relevant geometry at any tangentialposition along the length of the ski. In general, the term “curvature”is defined as the reciprocal of the local radius of a curvilinear form.The plane within which curvature is measured is dependent upon thenature of the particular curvilinear form of interest. For example, inthe context of bending deformations, flexural curvatures (bendingcurvatures) are measured within the local v-n plane of the ski.Similarly, camber and/or rocker curvatures are also measured within thelocal v-n plane of the ski. Conversely, the local curvature of thesidecut profile of the ski would be measured within the local v-b planeof the ski.

Although the present descriptions herein do make extensive use ofcurvatures in order to represent curvilinear forms, this is not intendedto be restrictive. For example, some embodiments could be formulatedbased upon direct three-dimensional representations of curvilinear formsin Cartesian coordinates. A reader who is skilled in the art willappreciate that a smooth and continuous curvilinear geometry can berepresented at various levels of differentiation and/or integrationwithout tarnishing the fidelity of the representation of saidcurvilinear geometry.

The present descriptions herein generally focus on two types ofstructural deformations: bending (also referred to as flexure) andtorsion (also referred to as twisting). In general, bending momentsimposed upon a ski (or other gliding board) will refer to moments thatact about the local b orientation vector of the ski, thus causingflexural curvatures (bending curvatures) that exist within the local v-nplane of the ski. Conversely, torsion moments (otherwise referred to astwisting moments, or torques) imposed upon a ski (or other glidingboard) will generally refer to moments that act about the local vorientation vector of the ski, thus causing torsional deformations(rotations) about the local v orientation vector of the ski.

The present descriptions herein also refer to total curvatures, whichrepresent the curvature of the overall deformed shape of the ski. Assuch, total curvatures represent the summation of camber curvatures,rocker curvatures, and flexural curvatures, wherein all of saidcurvatures exist within the local v-n plane of the ski.

A sign convention for curvatures is defined, wherein a positivecurvature is one that embodies a concave upward (convex downward)geometry within the local v-n plane of the ski; as such, positivecurvatures exhibit concavity that points in the same general directionas the local n orientation vector. A sign convention for bending momentsis defined, wherein a positive bending moment is one that will induce apositive flexural curvature.

The “flexural neutral axis” of the ski is defined as the elevationwithin the thickness of the ski at which bending stresses and bendingstrains are equal to zero.

The present descriptions herein will generally use the terms “flexural”and “bending” interchangeably.

In addition, the present descriptions herein will generally use the term“stiffness” in reference to a quantitative measurement of a resistanceto deformations and/or deflections; as such, the term “stiffness” isintended to have essentially the same meaning as “rigidity”. Conversely,the term “compliance” is used herein to denote the reciprocal ofstiffness; as such, a structure that exhibits relatively high stiffnessis one that exhibits relatively low compliance.

The term “camber curvature” is generally used herein to denote anegative curvature that exists in the absence of any applied loads orotherwise induced deformations. Conversely, the terms “rocker” and/or“reverse-camber” are generally used to denote a positive curvature thatexists in the absence of any applied loads or otherwise induceddeformations.

The terms “camber elevation profile” and “camber curvature profile” aregenerally used herein to denote the overall geometry of the base contourof the ski in the absence of any applied loads or otherwise induceddeformations; as such, although these terms do not explicitly includethe term “rocker”, the camber curvature profile of a ski may generallyrefer to an assembly (or summation) of both camber curvatures and rockercurvatures. In some cases, the present descriptions herein will moreexplicitly denote the overall geometry of the base contour of the ski asthe “camber and rocker” elevation profile or the “CamRock” elevationprofile; however, it will not always be necessary or practical to makethis explicit distinction.

The present descriptions herein use the term “spring-back” in referenceto deformations that the ski will exhibit upon removal from themanufacturing tooling (mould) as a result of residual internal strainenergy that is stored within each constituent of the ski duringfabrication. The term “elastic spring-back” is used herein in referenceto spring-back that is caused by strain energy that is stored in the skidue to the mechanical deformations that are imposed upon eachconstituent of the ski during fabrication. Conversely, the term“thermo-mechanical spring-back” is used herein in reference tospring-back that is caused by thermal strain energy that is stored inthe ski due to the change in temperature that is imposed upon eachconstituent of the ski during fabrication.

The “snow penetration depth” is defined herein as the distance betweenthe surface of the snow and lowest point of the ski (the edge that iscurrently engaged with the snow), measured perpendicular to the surfaceof the snow. The “local snow penetration depth” denotes the penetrationdepth at a specific tangential coordinate along the length of the ski.The “snow penetration depth profile” defines the local snow penetrationdepth at any position along the length of the ski under design pointconditions. The “global penetration depth” of the ski is defined hereinas the distance between the mid-base point of the ski and the surface ofthe snow, measured perpendicular to the surface of the snow. The “pitchangle” of the ski is defined herein as the angle between the surface ofthe snow and the v orientation vector that is positioned at the mid-basepoint of the ski, wherein said pitch angle is measured about an axisthat lies parallel to the surface of the snow and perpendicular to thelongitudinal axis of the ski.

While a broad variety of skiing styles and corresponding ski typesexist, racing and on-piste carving skiing techniques typically involvean athlete undertaking to execute carved turns. Carved turns aretypically carried out on prepared skiing trails (pistes) that arecovered with groomed snow, wherein said groomed snow has been packed(consolidated) and/or sintered, thus yielding a smooth and firm surfaceof snow. A carved turn can be characterized as a turn that is executedin a manner that minimizes the quantity of snow that is displaced fromits resting position on the piste, thus minimizing the drag (snowresistance) that is imposed upon the ski during the turn.

It is convenient to define a “snow trace” as the instantaneouscurvilinear geometry of the effective line of contact between the skiand the snow. In essence, the snow trace represents the effective lineof contact between the ski and the snow when said ski isquasi-statically pressed into the snow under comparable loadingconditions (comparable edging angle, loading, environmental conditions,etc.) to those which would be present during a carved turn, wherein saidski is not actually sliding along this effective line of contact. Insome example embodiments, this instantaneous effective line of contact(the snow trace) represents the line along which snow loads are assumedto be acting at each longitudinal position along the length of the ski.The term “snow trace profile” herein denotes the curvilinear geometry ofthis snow trace. In a dynamic sense, the snow trace profile representsthe instantaneous geometry of the groove or trench that is created bythe ski during a carved turn, wherein said groove or trench is onlyobserved within the vicinity of the instantaneous position of the ski.

A carved turn is initiated when a ski engages with the snow at somenon-zero edging angle, the ski de-cambers onto the surface of the snow,and the edge of the ski cuts a curvilinear groove into the surface ofthe snow that approximately matches the curvilinear geometry of the snowtrace. For convenience, the term “carved groove” herein denotes thegroove that is cut by the ski during the carved turn. If the ski slidesthrough the carved groove cleanly without significantly disturbing thesnow on either side of the groove, and while minimizing the width of thecarved groove, then the ski is said to have executed a carved turn. Assuch, in order for an ideal carved turn to be achieved, theinstantaneous velocity vector of any point along the length of the skishould be approximately tangential to the snow trace profile near thelongitudinal position of said point; this is particularly true in thevicinity of the afterbody of the ski. Conversely, if the instantaneousvelocity vector of some point along the length of the ski (particularlya point within the vicinity of the afterbody of the ski) is notapproximately tangential to the snow trace profile near the longitudinalposition of said point, then the ski will tend to disturb a largerquantity of snow, thus resulting in the creation of a relatively widecarved groove, and resulting in a skidded (non-carved) turn.

For any snow trace profile, there may exist a corresponding globalcurvature profile. In some example embodiments, this global curvatureprofile embodies a curvilinear form that can be determined by projectingthe curvilinear geometry of the snow trace profile onto a globalcurvature projection plane. In some example embodiments, this globalcurvature projection plane constitutes an oblique projection plane thatis inclined with respect to the plane on which the snow trace profileexists (the snow surface). In some example embodiments, the globalcurvature projection plane constitutes a nominal plane of symmetry thatis oriented approximately parallel to the longitudinal axis of the skiand approximately perpendicular to the base surface of the ski, when theski is oriented at the edging angle that corresponds to the relevantsnow trace profile.

Some embodiments described herein make use of various functions,subroutines, and other software algorithms that utilize numerical searchoperations that employ a bisection method (herein referred to as“bracketed bisection search algorithms”). While bracketed bisectionsearch algorithms are described herein due to their robust convergencecapabilities, some embodiments may use a broad variety of other searchmethods that may also be conducive for solving each of the numericalsearch problems that are discussed herein. Such alternative searchmethods might include, but are not limited to: the golden-section searchmethod, the secant method, the false position method, and various othernumerical search methods. Although the selected search method could havea significant effect upon the rate and robustness of convergence upon asolution, the specific search method that is employed may not beregarded as critical in at least some embodiments. As such, bracketedbisection search algorithms are employed in the descriptions herein forillustrative purposes only, and their use should not be regarded asrestrictive.

An aspect of this description relates to the creation of alpine skidesigns, such as designs for skis meant for racing and on-piste carvingthat are optimized for an individual athlete in the context ofspecifically defined conditions and performance-based parameters.

An aspect of this description relates to a method of identifying the skiequipment suitable for an athlete, as an alternative to or complementinga guess and check method. An aspect of this description takes intoaccount sidecut profile, camber elevation profile, flexural stiffnessprofile, and torsional stiffness profile in creating ski equipment, suchas to create a ski which will exhibit an ideal deformed shape and snowpressure distribution under the selected optimization parameters.

An aspect of this description relates to a more scientific andrepeatable approach to identifying the ideal skis for an athlete, to getthe most out of each athlete's potential without resorting to aguess-and-check approach. An aspect of this description relates togenerating an ideal ski design based upon meaningful performance-basedparameters, while retaining direct independent control over eachparameter.

An aspect of this description relates to designing based uponperformance-based parameters, so that varying any one parameter onlysubstantially influences the relevant performance characteristic. Anaspect of this description can be contrasted to designing based upon skidesign characteristics (such as stiffness, camber, sidecut, etc.);designing based upon ski design characteristics means that varying oneparameter will likely influence a multitude of performancecharacteristics. For example, an athlete may ask for a ski design to berevised such that one or more specific performance characteristics arechanged. Embodiments of the present system and method may facilitatesuch a design change without substantially changing other performancecharacteristics, and without tarnishing the overall harmony of the skidesign such as the relationship between sidecut, stiffness, and camber.

An aspect of this description relates to a system and method to buildnumerous versions of a single ski design (a “family” of ski designs)which are each optimized for a design-specific set of conditions (snowhardness, temperature, etc.).

An aspect of this description relates to a system and method to converta ski design that works well for one athlete to another athlete byaltering only the athlete-specific parameters while keeping all otherparameters constant. For example, if athlete A is pleased with his/herskis, and athlete B is having difficulty finding a ski design that worksfor him/her, it is possible to take athlete A's ski design, and alteronly the optimization parameters that are athlete specific (weight, bootsize, binding release setting, etc.) in order to enable athlete B tohave access to a nearly identical on-snow experience that athlete A has.In this example, the resulting ski design can then be further refined inorder to home-in on athlete B's personal preferences. In this example,this approach may serve as a rapid method of establishing a ski designthat is very close to one that would please athlete B.

In some embodiments, if an athlete prefers one particular binding and/orplate system, the system or method can generate a ski design that isoptimized to work with the binding/plate system of choice.

In some embodiments, a system and method allow a user to test the effectof using new materials and/or laminate architectures in the ski withoutaltering the stiffness of the ski. For example, it may be possible tocompare two ski designs that are identical in nearly every way butcomprise different core materials.

An aspect of this description relates to experimenting with geometricski design parameters (such as shovel design, early-rise, taper, nominalradius, etc.), without damaging the harmonious relationship between allother ski characteristics.

Referring now to FIG. 2, shown therein is a system 2000 for generating adesign of a ski. Ski design system 2000 includes a processing unit 2100,a system memory 2200, an output interface 2500, an input device 2400,and a network interface 2600, which are interconnected across a systembus or network 2300.

Although ski design system 2000 is shown as one component in FIG. 2, insome embodiments, the ski design system 2000 can be provided with one ormore servers distributed over a wide geographic area and connected via anetwork.

Each of the components of ski design system 2000 may be combined into afewer number of components or may be separated into further components.Each of the components of ski design system 2000 may be implemented insoftware or hardware, or a combination of software and hardware.

The processing unit 2100 may be any suitable processor, controller, ordigital signal processor that provides sufficient processing powerdepending on the configuration, purposes, and requirements of the skidesign system 2000. In some embodiments, the processing unit 2100 caninclude more than one processor with each processor being configured toperform different dedicated tasks. The processing unit 2100 controls theoperation of the ski design system 2000. For example, the processingunit 2100 can receive a plurality of inputs related to a particularathlete or condition and generate, in accordance with the methodsdisclosed herein, from the inputs, a design for a ski.

The system memory 2200 can include Random-Access Memory (RAM), Read-OnlyMemory (ROM), one or more hard drives, one or more flash drives, and/orsome other suitable data storage elements such as disk drives, etc. Forexample, the system memory 2200 can include a memory on which one ormore databases or file system(s) are stored. The database(s) can storeinformation related to generating a ski design, such as, but not limitedto, athlete parameters, technique and performance parameters, nominalgeometric parameters, ski construction, environmental conditions,binding and plate characteristics, etc. In some embodiments systemmemory 2200 may include an operating system, application programs, andprogram data.

The ski design system 2000 is networked through the network interface2600 and communicatively coupled to the remote device 2700. The networkinterface 2600 may be any interface that enables the ski design system2000 to communicate with other devices and systems. In some embodiments,the network interface 2600 can include at least one of a serial port, aparallel port, or a Universal Serial Bus (USB) port. The networkinterface 2600 may also include at least one of an Internet, Local AreaNetwork (LAN), Ethernet, Firewire, modem, or digital subscriber lineconnection. Various combinations of these elements may be incorporatedwithin the network interface 2600.

The remote device 2700 may be any networked device operable to connectto the network interface 2600. A networked device is a device capable ofcommunicating with other devices through a network. A networked devicemay couple to a network through a wired or wireless connection. Althoughonly one remote device 2700 is shown in FIG. 2, there may be multipleremote devices in communication with the ski design system 2000 via thenetwork interface 2600. The remote devices 2700 can be distributed overa wide geographic area. The remote devices 2700 may include at least aprocessor and memory, and may be an electronic tablet device, a personalcomputer, workstation, server, portable computer, mobile device,personal digital assistant, laptop, smart phone, Wireless ApplicationProtocol (WAP) phone, an interactive television, video displayterminals, gaming consoles, and/or portable electronic devices, or anycombination of these. In some embodiments, the remote devices 2700 maybe a laptop or a smartphone device equipped with a network adapter forconnecting to the Internet.

A user may interface with the ski design system 2000 through inputdevices 2400 to provide information and instructions. For example, inputdevices 2400 may include a mouse, a keyboard, a touch screen, athumbwheel, a track-pad, a track-ball, a card-reader, voice recognitionsoftware and the like depending on the requirements and implementationof the ski design system 2000.

A user may receive information from the ski design system 2000 throughoutput interface 2500 or remote device 2700. For example, a user mayreceive presentations, media files, items, and/or other content relatedto a generated ski design. The output interface 2500 may be a monitor, aspeaker, a network interface, or any other suitable interface dependingon the requirements and implementation of the ski design system 2000.

System 2000 can request inputs, such as from one or more user-operatedinput devices 2400, a storage system such as system memory 2200, and/ora remote device such as remote device 2700. For example, system 2000 mayrequest athlete parameters, technique and performance parameters,nominal geometric parameters, ski construction, environmentalconditions, and/or binding and plate characteristics, etc. System 2000can then generate a design for a ski based on the requested inputs. Aswill be explained below with reference to FIGS. 12-14 and 16-17, system2000 can apply various methods to generate the ski design.

Using system 2000, any one or more optimization parameters can bealtered independently, and the influence of those parametric changeswill be realized in the resulting ski design that is generated by theprocess. System 2000 can generate ideal ski designs based uponmeaningful performance parameters, while retaining direct independentcontrol over each parameter.

Using system 2000, designing based upon performance-based parametersmeans that varying one parameter may only influence the relevantperformance characteristic. Conversely, designing based upon ski designcharacteristics (such as stiffness, camber, sidecut, etc.) means thatvarying any one parameter would likely influence a multitude ofperformance characteristics. For example, suppose an athlete wants a skito exhibit more “pop” or “energy”. In this case, an engineer mightdecide to increase the ski's camber. However, this design change wouldnot only increase “pop” or “energy”, it would also increase the apparentstiffness of the ski, it would change the optimum edging angle of theski, and as a result, it would alter the conditions under which the skiexhibits optimum carving performance. Conversely, system 2000 can allowthe engineer to change a single performance characteristic (such as onethat would increase “pop” or “energy”) while minimizing any influencethat this change has upon other unrelated performance characteristics.In the present case of an athlete asking for more “pop”, the user canstipulate changes to the input parameters that would yield an increasein the ski's camber, and system 2000 would automatically make a seriesof additional corresponding design changes (likely including subtlechanges to the camber elevation profile, the stiffness profiles, and thesidecut profile) in order to isolate the desired performance changewhile retaining all other performance characteristics as much aspossible. The ski design system 2000 can also ensure that the conditionsunder which the ski exhibits optimum carving performance (the designpoint) are unchanged, and the harmonious relationship between theflexural stiffness profile, torsional stiffness profile, camberelevation profile, and sidecut profile (width profile) is preserved.

Another example that helps to illustrate the benefits of designing basedon performance based parameters is a scenario where an athlete tends toadopt an athletic stance with his/her weight concentrated relatively farforward on the ski, said athlete perhaps concentrating his/her weightcloser to his/her toes rather than near the mid-sole of his/her boots.In this case, it may be desirable to move the perceived “sweet spot”further forward on the ski, wherein said sweet spot represents thelongitudinal position of the load application point that yields the bestpossible snow load profile. An engineer employing conventional designtechniques may choose to achieve this change in sweet spot position byincreasing the flexural stiffness of the ski's forebody and decreasingthe flexural stiffness of the ski's afterbody. However, this designchange would not only change the location of the sweet spot; it wouldalso change the shape of the snow load profile and the snow tracegeometry, wherein said changed snow load profile may stray further froma desired snow load profile, and wherein said changed snow tracegeometry may stray further from a desired snow trace geometry.Conversely, system 2000 can allow the engineer to alter the desiredposition of the sweet spot of the ski while maintaining the harmoniousrelationship between the various design characteristics of the ski.Consequently, employing system 2000 to change the position of the sweetspot of the ski would likely result in a slightly skewed version of thesnow load profile that existed prior to this change, and the inclinationof this snow load profile would likely be altered slightly in order toalign the centroid of the snow load profile with the position of thesweet spot; however, the general characteristics of the snow loadprofile would be left substantially unchanged. In addition, employingsystem 2000 to change the position of the sweet spot of the ski wouldlikely result in only a negligible (likely imperceptible) change to theresulting snow trace geometry that is generated by the altered skidesign.

Referring now to FIG. 1, shown therein is an example method 1000 foroperating the ski design system 2000 to generate a ski design. Method1000 begins at 1100, where the ski design system 2000 receives a set ofinput athlete parameters. Input athlete parameters may include athleteweight, boot size, binding release settings, and binding mountingposition.

At 1200, the ski design system 2000 receives technique and performanceparameters. Technique and performance parameters may include therelationship between angulation and inclination of the athlete, weightdistribution between left and right feet, optimum edging angle, and foreand aft center of gravity position.

At 1300, the ski design system 2000 receives nominal geometricparameters.

Nominal geometric parameters may include length of effective baseregion, nominal sidecut radius, taper, and shovel and tail geometry.

At 1400, the ski design system 2000 receives ski constructionparameters. Ski construction parameters may include material mechanicalproperties, material physical properties, and laminate architecture.

At 1500 the ski design system 2000 receives environmental conditionparameters. Environmental condition parameters may include aconstitutive snow model and one or more temperatures.

At 1600 the ski design system 2000 receives binding and platecharacteristic parameters. Binding and plate characteristic parametersmay include binding dimensions, plate dimensions, plate stiffness, toeand heel clamping system characteristics, brake pedal characteristics,and forward pressure system characteristics.

At 1700, the ski design system 2000 generates a ski design. In someembodiments, the ski design system 2000 may generate the geometric datanecessary to construct the skis in accordance with a target design (forexample: geometry of each layer, mould geometry, core thickness profile,etc.). In some embodiments, the ski design system 2000 may generate asummary of the expected mechanical properties of the as-built ski designsuch that the accuracy of the construction can be verified once the skishave been built. For example, the ski design system 2000 may generateoutput files, which can be written to a directory in which the inputfiles were stored. The output data may include: overall geometry anddimensions of the ski design, geometry and dimensions of each of theconstituents of the ski design (as necessary to construct the ski), thetargeted camber elevation profile of the ski, geometry of the mould(tooling) that would be required in order to achieve the targeted camberelevation profile of the ski, and the expected mechanical performance(flexural and torsional stiffness profiles) of the ski. In addition, skidesign system 2000 may generate a series of plots that would help a userto assess the general characteristics of the final ski design,including: the expected camber elevation profile of the ski at varioustemperatures that were provided by the user, the expected flexuralstiffness profile of the ski, and the expected snow pressuredistribution over the length of the ski under design point conditions.

The ski design system 2000 can generate the ski design based on theinputs received at steps 1100-1600. The ski design system 2000 cangenerate the ski design using various methods. For ease of exposition,reference will now be made to FIGS. 16-17 and 12-14, to illustrateexample methods applied by the ski design system 2000 to generate a skidesign.

Referring now to FIG. 16, shown therein is an example method 100 foroperating the ski design system 2000 to generate a ski design. Method100 begins at 100A, where the ski design system 2000 defines parametersthat describe a desired carved turn of the ski. For example, the skidesign system 2000 can determine a nominal edging angle of the ski, andcan employ all available information to determine an athlete loadprofile, wherein said athlete load profile represents the total athleteload that can be assumed to be applied by the athlete to the ski duringthe carved turn. In some example embodiments, the athlete load profilemay include the total magnitude of the total athlete load that can beassumed to be applied by the athlete to the ski during the carved turn.In some example embodiments, athlete load profile may include thedirection of the total athlete load that can be assumed to be applied bythe athlete to the ski during the carved turn. In some exampleembodiments, athlete load profile may include the position of thecentroid of the total athlete load that can be assumed to be applied bythe athlete to the ski during the carved turn. For example, the skidesign system 2000 may determine the athlete load profile based at leaston an edging angle of the ski, an inclination angle of the athlete, amass of the athlete, and a weight distribution of the athlete. In someexample embodiments, the ski design system 2000 may determine theathlete load profile based at least on an edging angle of the ski, aninclination angle of the athlete, an angle of inclination of the pisteon which the carved turn is to be executed, a mass of the athlete, and aweight distribution of the athlete between his/her left and right feet.In some example embodiments, a user may directly provide some or allaspects of the athlete load profile as an input, thus alleviating therequirement for the ski design system 2000 to calculate said aspects ofthe athlete load profile.

At 1008, the ski design system 2000 defines a desired global curvatureprofile. The desired global curvature profile corresponds to a desiredsnow trace profile for the desired carved turn. For example, the skidesign system 2000 can determine a desired groove or trench (hereinreferred to as a snow trace) that the ski would create in the snow ifsaid ski were to be quasi-statically pressed into the snow undercomparable loading conditions (comparable edging angle, loading,environmental conditions, etc.) to those which would be present duringthe carved turn. The term “snow trace profile” herein denotes thecurvilinear geometry of this snow trace. In a dynamic sense, the snowtrace profile represents the instantaneous geometry of the groove ortrench that is created by the ski during a carved turn, wherein saidgroove or trench is only observed within the vicinity of theinstantaneous position of the ski. In other words, the snow traceprofile represents the geometry of the instantaneous effective line ofcontact between the ski and the snow. In some example embodiments, thisinstantaneous effective line of contact represents the line along whichsnow loads are assumed to be acting at each longitudinal position alongthe length of the ski. The ski design system 2000 may define the snowtrace profile under circumstances wherein the ski is not actuallysliding through the grove that is created within the snow. It is worthnoting that the aforementioned snow trace profile may differ slightlyfrom the geometry of the groove or trench that would be created by a skiunder dynamic conditions during the carved turn (the carved groove);this is due, in part, to the fact that the forebody of the ski must cutthe groove, whereas the afterbody of the ski may simply pass through theresulting grove. However, in other example embodiments, the ski designsystem 2000 may define the snow trace profile such that it is assumed tobe identical the geometry of the grove or trench that the ski would cutunder dynamic conditions as the carved turn is executed (the carvedgroove).

The desired snow trace profile can correspond to the desired globalcurvature profile. In some example embodiments, this desired globalcurvature profile can embody a curvilinear form that can be determinedby projecting the curvilinear geometry of the desired snow trace profileonto a global curvature projection plane. In some example embodiments,this global curvature projection plane constitutes a nominal plane ofsymmetry that is oriented approximately parallel to the longitudinalaxis of the ski and approximately perpendicular to the base surface ofthe ski, when the ski is oriented at an edging angle (the nominal edgingangle) that corresponds to the carved turn.

In some embodiments, the geometry of the desired snow trace profile maybe selected so as to minimize the quantity of snow that may be displacedby a ski and/or to maximize the engagement (or grip) of the ski with thesnow, and therefore to maximize the carving performance of a ski. Insome embodiments, the desired snow trace profile may correspond to anapproximately constant turning radius. For example, in some exampleembodiments that define the global curvature projection plane asconstituting a nominal plane of symmetry, a desired snow trace profilemay be defined such that it corresponds to an approximately constantturning radius; in such example embodiments, the corresponding desiredglobal curvature profile may be approximately representative of anelliptical arc (partial ellipse) geometry, because a constant radiuscircular arc (in this case, the snow trace profile) that is projectedonto an inclined plane (in this case, the nominal plane of symmetry)takes the form of an elliptical arc (partial ellipse). In someembodiments, the desired snow trace profile and the correspondingdesired global curvature profile can be determined based on nominal orinitial inputs provided by a user, such as a nominal width profile, anominal sidecut profile, and/or a ski edging angle.

At 100C, the ski design system 2000 generates a desired deformed shapeof the ski during the desired carved turn. The desired deformed shape ofthe ski can be defined at least by a desired total curvature profile,wherein the desired total curvature profile is initially set tocorrespond to the desired global curvature profile. For example, the skidesign system 2000 can assume a deformed shape (and corresponding totalcurvature profile) of the forebody of the ski that would generate a snowtrace approximately having the desired global curvature profile, and adeformed shape (and corresponding total curvature profile) of theafterbody of the ski that can approximately smoothly pass through theresulting carved groove. In some embodiments, the desired totalcurvature profile of the deformed ski may initially be set equal to thedesired global curvature profile. In some embodiments, the desired totalcurvature profile of the ski may be defined in a manner that initiallyignores the effects of torsional deformations.

In some embodiments, the desired total curvature profile may be subjectto modification during operations that are subsequently carried out bythe ski design system 2000.

In some embodiments, the ski design system 2000 can generate a desiredtotal curvature profile based at least on an initial width profile and aski edging angle. For example, the ski design system 2000 may determinea desired total curvature profile based on a nominal sidecut profile anda nominal edging angle received from a user.

The use of a total curvature profile here serves as a convenient meansof representing the overall deformed shape of the ski, while excludingany torsional deformations from this representation; however, thiscurvature-based representation should not be considered restrictive.Some alternative embodiments may employ other representations of theoverall deformed shape of the ski that exclude torsional deformations,such as a direct three-dimensional geometric representation of the bentbut untwisted ski in Cartesian coordinates.

In some example embodiments, it may be convenient to define a “totaldeformed geometry profile” that represents the three-dimensionalgeometry of the deformed shape of the ski, wherein said total deformedgeometry profile can be defined in terms of the total curvature profileof the ski and the torsional deformation profile of the ski. By thisdefinition, there may also exist a desired total deformed geometryprofile of the ski, wherein said desired total deformed geometry profilemay be defined in terms of the desired total curvature profile of theski and the desired torsional deformation profile of the ski.

At 100D, the ski design system 2000 generates a sidecut profile of theski. This sidecut profile may be subject to modification duringoperations that are subsequently carried out by the ski design system2000. For example, the ski design system 2000 can generate a sidecutprofile based on nominal inputs provided by the user. For example, auser may provide a nominal sidecut radius, and the ski design system2000 can generate a sidecut profile that corresponds to said sidecutradius. In an alternative embodiment, a user may provide inputparameters that describe a sidecut profile that is not of a constantradius arc, and the ski design system 2000 can generate a sidecutprofile that corresponds to said sidecut input parameters.

In some example embodiments, the ski design system 2000 can define adesired geometry of a snow penetration depth profile, wherein the snowpenetration depth profile represents the local snow penetration depthvalue at each position along the length of the ski, and the ski designsystem 2000 can then generate a sidecut profile based at least on thedesired global curvature profile, the desired geometry of the snowpenetration depth profile, and a ski edging angle. In some exampleembodiments, the ski design system 2000 can generate a sidecut profilebased at least on the desired total curvature profile of the deformedski during the carved turn, the desired geometry of the snow penetrationdepth profile, and a ski edging angle. In some example embodiments, thisdesired geometry of the snow penetration depth profile may represent thegeneral shape of the desired snow penetration depth profile, withoutexplicitly defining the position or orientation of this shape relativeto a datum. For example, the user may define a W-shaped desired geometryof a snow penetration depth profile, wherein said snow penetration depthprofile exhibits local maxima values in the vicinities of the shovel,the tail, and the mid-base regions of the ski; however, this W-shapedgeometry may be translated, rotated, or skewed when it comes time forthe ski design system 2000 to determine a resulting snow penetrationdepth profile. In some embodiments, the desired geometry of the snowpenetration depth profile may correspond to an approximately constantsnow penetration depth or may result in an approximately constant snowpressure. In some embodiments, the desired geometry of the snowpenetration depth profile may correspond to a snow penetration depththat approximately follows a linear function (a polynomial function ofdegree one or zero) over the length of the effective base region of theski. For example, the ski design system 2000 can determine a sidecutprofile that, during the carved turn, will generate a snow penetrationdepth profile having a geometry that is substantially similar and/orproportional to the desired geometry of the snow penetration depthprofile that was stipulated by the user. For example, if the userstipulates that the desired geometry of the snow penetration depthprofile is to follow a linear function, then the ski design system 2000can determine a sidecut profile that, during the carved turn, willgenerate a snow penetration depth profile that approximately follows alinear function. In the aforementioned example of a linear snowpenetration depth profile, if the athlete load profile has a centroidthat is rearward of the mid-base point, then the resulting linear snowpenetration depth profile will likely have a maximum local snowpenetration depth value at a position that is near the rearward-mostpoint of the effective base region of the ski.

In some embodiments, the ski design system 2000 can determine thesidecut profile based on an initial nominal sidecut profile of the skiand a ski edging angle. For example, the ski design system 2000 maydetermine a desired snow trace profile based at least on a nominalsidecut profile and a nominal edging angle received from a user, andthen the ski design system 2000 may determine a corresponding sidecutprofile such that the desired snow trace profile and the desiredgeometry of the snow penetration depth profile are simultaneouslyachieved at the stipulated nominal edging angle. In an alternativeexample, the user may directly stipulate the desired snow trace profile,and then the ski design system 2000 may determine a correspondingsidecut profile such that the desired snow trace profile and the desiredgeometry of the snow penetration depth profile are simultaneouslyachieved at the stipulated nominal edging angle.

At 100E, the ski design system 2000 generates a width profile of theski. This width profile may be subject to modification during operationsthat are subsequently carried out by the ski design system 2000. In someexample embodiments, the width profile may be determined as a functionof the sidecut profile, as well as initial inputs provided by a user,such as: the waist width of the ski, and the taper of the ski. In thecase of a ski that has left and right sidecut profiles that are notsymmetric about the longitudinal axis of the ski, it may be necessary toindependently determine both the left and right sidecut profiles of theski, and then calculate the width profile once both sidecut profileshave been determined.

At 100F, the ski design system 2000 generates a camber profile of theski. In some example embodiments, the camber profile is represented by acamber curvature profile, which defines the curvature of the camberprofile at each position along the length of the ski. In some exampleembodiments, the ski design system 2000 can determine a camber profilethat, during the carved turn, allows the ski to deform to the desiredtotal curvature profile and accordingly approximately achieves thedesired snow trace profile.

In some example embodiments, the ski design system 2000 may determinethe camber profile based on a total curvature profile and acorresponding load profile. For example, the ski design system 2000 candetermine the camber profile based on the desired deformed shape(wherein said desired deformed shape corresponds to the desired totalcurvature profile) of the ski during the carved turn and thecorresponding loads that are applied to the ski during the carved turn,as well as the corresponding edging angle. In some embodiments, the skidesign system 2000 may consider multiple total curvature profiles andcorresponding load profiles that correspond to different edging angles.

For example, the ski design system 2000 can determine a first totalcurvature profile and a first load profile that correspond to a firstedging angle. The ski design system 2000 can also determine a secondtotal curvature profile and a second load profile that correspond to asecond edging angle. The first and second total curvature profiles mayeach be determined in a manner similar to that employed in step 100C. Inalternative example embodiments, the first and second total curvatureprofiles may each constitute a single nominal total curvature value (forexample, a total curvature value that represents the total curvature atthe position of the mid-base point) that is representative of ahypothetical total curvature profile. The ski design system 2000 maythen determine a camber profile that facilitates a ski design that iscapable of satisfying each of said first and second total curvatureprofiles with its corresponding load profile. That is, the ski designsystem 2000 determines a camber profile for a ski design that canachieve the first total curvature profile during a carved turn at thefirst edging angle with the first load profile, and can also achieve thesecond total curvature profile during a carved turn at the second edgingangle with the second load profile. In some embodiments, the ski designsystem 2000 can determine the camber profile to also satisfy a zero loadprofile, resulting in a total curvature profile that is equal to thecamber curvature profile. In some embodiments, the ski design system2000 can determine the camber profile to also satisfy a zero edgingangle, corresponding to a zero total curvature profile.

In some embodiments, the camber profile may be subject to modificationduring operations that are subsequently carried out by the ski designsystem 2000.

At 100G, the ski design system 2000 generates at least one stiffnessdesign variable and at least one corresponding stiffness design variableprofile. At any position along the length of the ski, the flexuralstiffness and torsional stiffness of the ski may be dictated by therelationship between the local value of this at least one stiffnessdesign variable, the local width of the ski, the constructionarchitecture of the ski, and the mechanical properties of the materialsthat are present within the ski construction. As such, a resultingflexural stiffness profile and a resulting torsional stiffness profilemay both be dictated by at least the at least one stiffness designvariable profile, the width profile, the construction architecture ofthe ski, and the mechanical properties of the materials that are presentwithin the ski construction. For example, skis that employ sandwich-typeconstruction commonly employ core thickness as the primary stiffnessdesign variable. In other words, the thickness of the core of the skivaries over the length of the ski in order to achieve a desired flexuralstiffness value and torsional stiffness value at each position along thelength of the ski. In such embodiments, at least one of the at least onestiffness design variable profile is represented by a core thicknessprofile. In some example embodiments, the ski design system 2000 mayinitialize at least one of the at least one stiffness design variableprofile as a zero vector, which may then be populated with non-zerovalues at subsequent steps of the process.

In some example embodiments, a ski may be designed with a constructionthat is of substantially constant thickness over its entire length, butfeatures longitudinally oriented stiffening ribs on its upper surface;in this case, the stiffness design variable could take the form of atleast one characteristic of these stiffening ribs (such as the heightand/or width of each rib), such that the targeted flexural and/ortorsional stiffness at any position along the length of the ski can beachieved by adjusting said at least one variable characteristic of thestiffening ribs.

In some example embodiments, a ski may be designed such that it exhibitstwo or more zones over its length, wherein two or more of these zonesdiffer in their construction architecture and/or their dominantstiffness design variable.

At 100H, the ski design system 2000 generates the total load profilethat is acting on the ski. For example, the total load profile may bedefined to include a snow load profile and the athlete load profile. Insome example embodiments, the total load profile may be defined as thevector sum of a snow load profile and the athlete load profile.Generating the total load profile may include generating a desired snowpenetration depth profile.

In some example embodiments, the snow load profile may be calculatedbased at least on at least one snow property, wherein said snow loadprofile represents the loads that are applied to the ski by the snow. Insome embodiments, the ski design system 2000 may determine the snow loadprofile based at least on the athlete load profile and at least one snowproperty. Various snow models (for example, constitutive materialmodels), which will be discussed in further detail below, can be used asthe at least one snow property. In some embodiments the ski designsystem 2000 may determine the snow load profile such that a totalmagnitude of the snow load profile is approximately equal to a totalmagnitude of a relevant component of the athlete load profile. That is,the ski design system 2000 may approximately balance the loads appliedby the snow with the loads applied by the athlete. For example, the skidesign system 2000 may determine a desired snow penetration depthprofile such that a total magnitude of the resulting snow load profileis approximately equal to a total magnitude of a relevant component ofthe athlete load profile. In addition, in some embodiments, the skidesign system 2000 may determine the snow load profile such that thecentroid of the snow load profile is positioned approximately coincidentwith the centroid of a relevant component of the athlete load profile.For example, the ski design system 2000 may determine a desired snowpenetration depth profile such that the centroid of the resulting snowload profile is positioned approximately coincident with a relevantcomponent of the centroid of the athlete load profile.

A reader who is skilled in the art will appreciate that a slightdiscrepancy may exist between the total magnitude of the athlete loadprofile and the total magnitude of the relevant component of the snowload profile, wherein said discrepancy may be caused, in part, byinertial loads that act upon the ski, the binding, the binding interfaceplate, and/or any other component that is not accounted for whendetermining the athlete load profile.

In some example embodiments, the ski design system 2000 may furtherdetermine a binding load profile as a part of determining the total loadprofile, wherein said binding load profile represents the various loadsthat the binding system imposes upon the ski. For example, the skidesign system 2000 may consider the various loads applied by the bindingsystem to the ski. These binding loads may include loads that accountfor the mechanical response of the various components of the bindingsystem, and how these components interact with the boot that is to berestrained by said binding. In some embodiments, the binding loads mayaccount for changes in the binding loads caused by articulations of thevarious mechanical components of the binding system, wherein saidarticulations may be caused, in part, by the deformation of the ski. Insome example embodiments, the binding load profile may include theeffects of a binding interface plate that is mounted between the ski andone or more components of the binding system; in such embodiments, thebinding load profile may include loads that are imposed upon the ski asa result of deformations of said binding interface plate, and theresulting structural response of said deformed binding interface plate.

In some example embodiments, the calculated binding load profile mayinclude the effects of the athlete load profile; in some suchembodiments, the total load profile may include at least the snow loadprofile and the binding load profile. In some such example embodiments,the total load profile may be defined as the vector sum of the snow loadprofile and the binding load profile.

In some example embodiments, the calculated binding load profile may notinclude the effects of the athlete load profile; in some suchembodiments, the total load profile may include at least the snow loadprofile, the athlete load profile, and the binding load profile. In somesuch example embodiments, the total load profile may be defined as thevector sum of the snow load profile, the athlete load profile, and thebinding load profile.

In some embodiments, wherein generating the total load profile includesdetermining a desired snow penetration depth profile, the desired snowpenetration depth profile is stored for subsequent use in calculationsthat require a desired snow penetration depth profile.

At 100I, the ski design system 2000 modifies at least the width profile,the sidecut profile, and at least one of the at least one stiffnessdesign variable profile at least once. For example, the ski designsystem 2000 may perform iterative or recursive methods to modify thewidth profile, the sidecut profile, and at least one of the at least onestiffness design variable profile. For ease of exposition, an exampleiterative method will now be described with reference to FIG. 17.

Referring now to FIG. 17, shown therein is an example method 200(corresponding to 100I of example method 100 shown in FIG. 16) ofoperating the ski design system 2000 to modify at least the widthprofile, the sidecut profile, and at least one of the at least onestiffness design variable profile. Method 200 begins at 200A, where theski design system 2000 calculates the desired flexural stiffness profilethat is necessary in order to achieve the desired total curvatureprofile of the deformed ski during the carved turn. For example, sincesystem 2000 has already determined a camber curvature profile and adesired total curvature profile, it can determine the desired flexuralcurvature profile as the difference between these profiles. System 2000may then calculate the desired flexural stiffness profile based on thedesired flexural curvature profile and the total load profile. The skidesign system may determine the desired stiffness profile based on thecamber profile, the desired flexural deformation profile, and the totalload profile of the ski. For example, the ski design system 2000 maydetermine a desired stiffness profile by considering the non-deformedshape of the ski, the deformed shape of the ski, and the loads appliedto the ski to cause the ski to deform between the non-deformed anddeformed shapes.

At 200B, the ski design system 2000 modifies at least one of the atleast one stiffness design variable profile, as necessary, such that aresulting flexural stiffness profile is approximately equal to thedesired flexural stiffness profile. In cases where the at least onestiffness design variable profile yields a resulting flexural stiffnessprofile that is equal to the desired flexural stiffness profile, the atleast one stiffness design variable profile is said to have “achieved”the desired flexural stiffness profile. For example, if a core thicknessprofile is employed as the stiffness design variable, then the skidesign system 2000 may determine a desired flexural stiffness profileand determine a core thickness profile that can approximately achievethe desired flexural stiffness profile.

In some embodiments, the values that populate at least one of the atleast one stiffness design variable profile may be limited to anallowable range having upper and/or lower bounds that are dictated bypractical considerations. For example, in some embodiments that employcore thickness as one of the at least one stiffness design variable, thecore thickness profile may be limited to a predetermined thicknessrange. For example, there may be practical limitations to a ski design,which result in a maximum or minimum core thickness. Accordingly, themodified core thickness profile may, in some cases, fail to achieve thedesired flexural stiffness profile. Thus, the ski design system 2000 maydetermine a core thickness profile that is as close as possible toapproximately achieving the desired flexural stiffness profile, whileremaining within the predetermined core thickness range. It will beunderstood that, in such embodiments, the ski design system 2000 canstill be said to approximately achieve the desired total curvatureprofile.

In some embodiments, the ski design system 2000 may include an extrastep (not shown) wherein the camber profile is modified based at leaston the at least one stiffness design variable profile in order toapproximately achieve the desired total curvature profile. For example,the ski design system 2000 may modify the camber profile locally inorder to compensate the ski in regions where the at least one stiffnessdesign variable profile (such as a core thickness profile) fails toachieve the desired flexural stiffness profile, and consequently the skifails to achieve the desired total curvature profile. In someembodiments, the ski design system 2000 can determine a resulting totalcurvature profile of the ski and compare the resulting total curvatureprofile to the desired total curvature profile to determine the requiredadjustments to the camber profile. For example, the ski design system2000 can determine the resulting flexural stiffness profile of the skibased at least on the at least one stiffness design variable profile,the width profile, the construction architecture of the ski, and atleast one material property of the ski (e.g., an elastic modulus of thecore). The ski design system 2000 can then determine the resultingflexural curvature profile of the ski based on the resulting flexuralstiffness profile and the total load profile acting on the ski. The skidesign system 2000 can then determine the resulting total curvatureprofile of the ski based on the resulting flexural deformation and thecurrently assumed camber profile. The ski design system 2000 can thenmodify the camber profile to compensate for regions where the resultingtotal curvature profile differs from the desired total curvatureprofile, and thus approximately achieve the desired total curvatureprofile. For example, the ski design system 2000 may add camber wherethe ski is insufficiently stiff, or add rocker where the ski isexcessively stiff.

In some embodiments, the ski design system 2000 may employ regions oftailored transverse shear compliance in order to compensate for regionsof excessive flexural stiffness. For example, the ski design system 2000can determine a resulting total curvature profile of the ski and comparethe resulting total curvature profile to the desired total curvatureprofile in order to determine the magnitudes and locations ofdeficiencies in the flexural compliance of the ski. In such regions ofdeficient flexural compliance, ski design system 2000 can add layers ofmaterials having low shear stiffness (such as thin foils of elastomericmaterials) in order to locally generate inclement transverse shearcompliance (shear compliance within the local v-n plane of the ski) thatcould mimic the effects of flexural compliance, thus compensating fordeficiencies in the actual flexural compliance of the ski.

At 200C, the ski design system 2000 calculates a torsional deformationprofile of the ski during the carved turn. For example, in someembodiments, the ski design system 2000 may determine a resultingtorsional stiffness profile of the ski based on the at least onestiffness design variable profile, the width profile, the constructionarchitecture of the ski, and at least one material property of the ski(e.g., an elastic modulus of the core). In some embodiments, the skidesign system 2000 may then determine a torsional deformation profilebased on the total load profile that is assumed to be acting on the skiand the resulting torsional stiffness profile of the ski. For example,the ski design system 2000 can determine the various loads applied tothe ski during the carved turn and determine the deformation of the skicaused by the corresponding torsion moments.

At 200D, the ski design system 2000 modifies the desired total curvatureprofile of the deformed ski based at least on a torsional deformationprofile in order to achieve a resulting global curvature profile that isapproximately equal to the desired global curvature profile. Forexample, the ski design system 2000 may determine a resulting deformedshape of the ski based at least on the currently assumed desired totalcurvature profile and the torsional deformation profile of the ski. Asdiscussed above, the initial desired total curvature profile of thedeformed ski determined at step 100C may not have taken torsionaldeformations into consideration.

In some embodiments, the ski design system 2000 may determine aresulting global curvature profile of the ski during the carved turn andcompare the resulting global curvature profile to the desired globalcurvature profile to modify the desired total curvature profile. Forexample, the ski design system 2000 may determine a resulting snow traceprofile of the ski based at least on the resulting deformed shape of theski, wherein said resulting deformed shape of the ski includes bothflexural deformations and torsional deformations, and calculate theresulting global curvature profile that corresponds to said resultingsnow trace profile. The ski design system 2000 may then modify thedesired total curvature profile based at least on the comparison betweenthe resulting global curvature profile and the desired global curvatureprofile. For example, the ski design system 2000 may then adjust thedesired total curvature profile to compensate for regions where theresulting global curvature profile is different from the desired globalcurvature profile due, in part, to torsional deformations.

In some embodiments, the ski design system 2000 may establish aresulting deformed shape of the ski based at least on a total curvatureprofile and a torsional deformation profile. In some embodiments, thisresulting deformed shape of the ski may be based at least on the desiredtotal curvature profile and the torsional deformation profile. The skidesign system 2000 can identify a desired snow penetration depthprofile, which may have been calculated during a previous operation. Insome example embodiments, this desired snow penetration depth profilemay have been calculated at least prior to carrying out the most recentcalculation of a torsional deformation profile (200C) (for example, atstep 100H). In some example embodiments, the desired snow penetrationdepth profile may have been calculated during a previous cycle throughthe various operations present within method 200. The ski design system2000 may then establish a position and an orientation of the resultingdeformed shape of the ski, wherein said position and orientationcorrespond to the identified desired snow penetration depth profile. Forexample, the ski design system 2000 may identify at least one anchorpoint along the length of the ski; at each of said at least one anchorpoint, the ski design system 2000 may then query the identified desiredsnow penetration depth profile in order to find a desired local snowpenetration depth value that corresponds to the position of said anchorpoint. The ski design system 2000 may then establish a position andorientation of the resulting deformed shape of the ski such that theresulting local snow penetration depth value at each of the at least oneanchor point is approximately equal to the corresponding desired localsnow penetration depth value for said anchor point. The ski designsystem 2000 may then determine a resulting snow trace profile thatcorresponds to the resulting deformed shape of the ski and theestablished position and orientation of said resulting deformed shape ofthe ski. The ski design system 2000 may then determine a resultingglobal curvature profile that corresponds to the resulting snow traceprofile. The ski design system 2000 may then compare the resultingglobal curvature profile to the desired global curvature profile. Theski design system 2000 may then modify the desired total curvatureprofile based at least on the comparison between the resulting globalcurvature profile and the desired global curvature profile.

In some example embodiments that employ at least one anchor point toestablish a position and orientation of the resulting deformed shape ofthe ski for the purpose of modifying the desired total curvature profileof the deformed ski, said at least one anchor point may be defined at aposition where the torsional deformation of the ski does not cause thelocal effective edging angle of the ski to differ from the userstipulated nominal edging angle of the carved turn. For example, the atleast one anchor point may be defined at the position of a mountingpoint of a boot-binding component, wherein said mounting point of aboot-binding component represents a longitudinal position at which theedging angle of the ski is explicitly defined and is not subject to theeffects of torsional deformations.

At 200E, the ski design system 2000 modifies the sidecut profile basedat least on a torsional deformation profile in order to achieve aresulting snow penetration depth profile that is approximately equal tothe desired snow penetration depth profile. For example, the ski designsystem 2000 can adjust the sidecut profile of the ski based on thetorsional deformations and the influence that these torsionaldeformations have upon the resulting snow penetration depth profile.

In some embodiments, the ski design system 2000 can modify the sidecutprofile such that the resulting snow penetration depth profile issubstantially unchanged relative to that which was calculated during theprevious cycle through the various operations present within method 200.

In some embodiments, the ski design system 2000 may establish aresulting deformed shape of the ski based at least on a total curvatureprofile and a torsional deformation profile. In some embodiments, thisresulting deformed shape of the ski may be based at least on the desiredtotal curvature profile and the torsional deformation profile. The skidesign system 2000 can identify a desired snow penetration depthprofile, which may have been calculated during a previous operation. Insome example embodiments, this desired snow penetration depth profilemay have been calculated at least prior to carrying out the most recentcalculation of a torsional deformation profile (200C). In some exampleembodiments, the desired snow penetration depth profile may have beencalculated during a previous cycle through the various operationspresent within method 200. The ski design system 2000 may then establisha position and an orientation of the resulting deformed shape of theski, wherein said position and orientation correspond to the identifieddesired snow penetration depth profile. For example, the ski designsystem 2000 may identify at least one anchor point along the length ofthe ski; at each of said at least one anchor point, the ski designsystem 2000 may then query the identified desired snow penetration depthprofile in order to find a desired local snow penetration depth valuethat corresponds to the position of said anchor point. The ski designsystem 2000 may then establish a position and orientation of theresulting deformed shape of the ski such that the resulting local snowpenetration depth value at each of the at least one anchor point isapproximately equal to the corresponding desired local snow penetrationdepth value for said anchor point. The ski design system 2000 can thenmodify the sidecut profile of the ski such that, at each position alongthe length of the deformed ski, the resulting local snow penetrationdepth value is approximately equal to the corresponding local snowpenetration depth value that is defined at that position by theidentified desired snow penetration depth profile. In other words, theski design system 2000 can modify the sidecut profile of the ski suchthat a resulting snow penetration depth profile is approximately equalto the identified desired snow penetration depth profile.

In some example embodiments that employ at least one anchor point toestablish a position and orientation of the resulting deformed shape ofthe ski for the purpose of modifying the sidecut profile, said at leastone anchor point may be defined at a position where the torsionaldeformation of the ski does not cause the local effective edging angleof the ski to differ from the user stipulated nominal edging angle ofthe carved turn. For example, the at least one anchor point may bedefined at the position of a mounting point of a boot-binding component,wherein said mounting point of a boot-binding component represents alongitudinal position at which the edging angle of the ski is explicitlydefined and is not subject to the effects of torsional deformations.

At 200F, the ski design system 2000 modifies the width profile of theski. In some example embodiments, the width profile may be determined asa function of the sidecut profile, as well as initial inputs provided bya user, such as: the waist width of the ski, and the taper of the ski.In the case of a ski that has left and right sidecut profiles that arenot symmetric about the longitudinal axis of the ski, it may benecessary to independently modify both the left and right sidecutprofiles of the ski, and then calculate the modified width profile onceboth sidecut profiles have been modified.

As shown in FIG. 17, method 200 can be repeated or iterated. That is,following step 200F, the ski design system 2000 may once again begin toperform step 200A. In some embodiments, the ski design system 2000 mayiteratively repeat steps 200A to 200F until a predetermined threshold isreached. For example, the ski design system 2000 may compare the widthprofile, the sidecut profile, and/or at least one of the at least onestiffness design variable profile from the previous iteration with thewidth profile, the sidecut profile, and/or at least one of the at leastone stiffness design variable profile from the current iteration. Theski design system 2000 may exit this loop when the comparison results ina difference that is less than the predetermined threshold.

In some example embodiments, modifying at least the width profile, thesidecut profile, and at least one of the at least one stiffness designvariable profile (corresponding to example method 200 shown in FIG. 17,and method 100I of example method 100 shown in FIG. 16) furthercomprises modifying the total load profile and the corresponding desiredsnow penetration depth profile, wherein said modification may be carriedout in accordance with example method 100H (shown in FIG. 16). Thismodification may improve the fidelity of the resulting ski designprocess because it will help to ensure that the total load profile actsin a manner that accounts for the deformed shape of the ski. Conversely,if the total load profile is only calculated once and then the deformedshape of the ski evolves during subsequent refinement of the design,then the assumed snow load profile (for example) will not actperpendicular to the base of the deformed ski. In some embodiments, step100H of method 100 can be executed within method 200, prior to step 200A(instead of, or in addition to, being executed within method 100).

Referring again to FIG. 16, at 100J, the ski design system 2000 definesthe design for the ski based at least on the at least one stiffnessdesign variable profile, the width profile, and the camber profile. Asdescribed above, the ski design system 2000 may, in some embodiments,output data relating to the geometry and dimensions of the ski, thegeometry and dimensions of each component of the ski, the geometry of amould for constructing the ski, and the expected performance parametersand mechanical characteristics of the ski.

In some embodiments, the ski design system 2000 can determine a mouldprofile for manufacturing the ski. For example, the ski design system2000 can determine a spring-back of the ski following removal from apress mould and generate a mould profile based on the spring-back andthe camber profile. The ski design system 2000 can determine thespring-back based on at least one material property of the ski. Forexample, the ski design system 2000 may determine a spring-back causedby residual thermal strain energy and/or residual elastic strain energy.The ski design system 2000 can then generate the mould profile based onthe camber profile and the determined spring-back. For example, thedesign system 2000 can generate a mould profile which compensates forthe spring-back and accordingly can be used to manufacture a ski thatmay accurately achieve the desired camber profile.

In some embodiments, the ski design system 2000 can generate a pluralityof discretized elements for the ski. In these embodiments, the skidesign system 2000 can perform steps 100A to 100J using the discretizedelements. For example, a desired total curvature profile may berepresented by a plurality of local desired total curvatures, whereineach total curvature corresponds to a discretized element. Similarly, atorsional deformation profile may be represented by a plurality of localtorsional deformations, wherein each torsional deformation correspondsto a discretized element. The sidecut profile, width profile, camberprofile, and the at least one stiffness design variable profile may alsoinclude local values that each correspond to a discretized element. Insome embodiments, the discretized elements may be beam elements eachhaving a deformed and/or non-deformed shape that can be represented by ahelical geometric formulation.

It will be appreciated that method 100 is one example of a method foroperating the ski design system 2000 to generate a ski design. Othermethods for generating a ski design using the ski design system 2000 arepossible. For example, reference will now be made to FIGS. 12-14, whichillustrate further examples of methods for generating a ski design usingthe ski design system 2000.

Referring now to FIGS. 12-14, shown therein are further example methods2010, 2020, and 2030 for operating the ski design system 2000 togenerate a ski design. As will be explained in further detail below,method 2010 corresponds to an initialization process; method 2020corresponds to an optimization loop that uses the results from theinitialization method 2010; and method 2030 corresponds to a process forpreparing design specifications based on the results from theoptimization method 2020.

As depicted in FIG. 12, method 2010 serves to initialize an optimizationproblem, which can then be iteratively solved for using method 2020.Method 2010 can be summarized by the following general sequence ofsteps: 2010 i receive input data; 2010A calculate the applied athleteloading; 2010B discretize the ski; 2010C initialize the local edgingangle profile at the design point, assuming zero torsional deformation;2010D initialize the total curvature profile at the design point,assuming zero torsional deformation; 2010E initialize the camberelevation profile, assuming zero rocker; 2010F initialize the widthprofile; 2010G calculate the mechanical and physical properties of eachmaterial at each temperature control point; 2010H calculate therelationship between the width, core thickness, and stiffness of the skiat the optimization temperature control point, in light of theuser-stipulated laminate architecture of the ski; 2010I offset allassumed curvature profiles and corresponding tangential coordinatesystems to the elevation of the flexural neutral axis; and 2010Jinitialize the geometry of the boot-binding system, in light of theassumed camber elevation profile.

As depicted in FIG. 12, method 2010 begins at 2010 i, where the skidesign system 2000 receives input data. For example, a user may provideinput data through a variety of text files that contain information forthe ski design system 2000 to carry out methods 2010, 2020, 2030. Theuser can point the ski design system 2000 to a directory in which thevarious input files reside, and the user can trigger the ski designsystem 2000 to commence method 2010.

Continuing with the above example, the ski design system 2000 canreceive one or more of the following inputs files as input data:INPUTS_User_Params, INPUTS_User_B inding_Params, INPUTS_Binding_Params,INPUTS_Engineering_Params, INPUTS_Snow_Model, INPUTS_Core_Materials,INPUTS_Core_Laminate, INPUTS_Laminate_Materials,INPUTS_Laminate_Architecture, INPUTSSublaminate_Plies,INPUTS_Solid_Component_Materials, INPUTS_Solid_Component_Geometry,INPUTS_Temperatures, CamRock_Offset, and Sidecut_Offset. It should benoted that the aforementioned collection of input files merelyrepresents a configuration employed by one example. Some embodiments mayamalgamate two or more of these input files into one larger input file.Some embodiments may organize the required input data differentlyamongst a different collection of input files. For example, some of thedata that is contained within the INPUTS_User_Params input file couldinstead be contained within the INPUTS_Engineering_Params. Thecollection of input files that is discussed herein is presented as anexample that illustrates the function of the ski design system 2000;however, the particular arrangement of the inputs should not beconsidered restrictive.

INPUTS_User_Params is a set of parameters pertaining to the athlete,his/her mass, his/her skiing technique, and a small collection ofhis/her preferred nominal ski characteristics. In order to optimize thedesign of a ski, it may be necessary to select a single scenario (designpoint) that embodies the conditions under which the optimization is tobe carried out. At this design point, the ski is placed on the snow at aspecific design point edging angle (θ_(e) _(DP) ), and the athleteadopts a design point inclination angle (θ_(i) _(DP) ). Consequently,the athlete also adopts a design point angulation angle that is definedas: θ_(a) _(DP) =θ_(e) _(DP) −θ_(i) _(DP) . The skiing technique of theathlete is defined in INPUTS_User_Params as the instantaneous stance ofthe athlete at the design point, including: angulation angle,inclination angle, and the position of the centre-of-mass of the athletealong the length of the ski. This targeted centre-of-mass positionrepresents the ideal position at which the athlete should concentratehis/her mass in order to replicate the conditions under which the designof the ski was optimized. This targeted centre-of-mass position may bethought of as the “sweet spot” of the ski, as it is the centre-of-massposition that will yield the best possible carving performance from theski.

In order to optimize the camber elevation profile of the ski, it may benecessary to define two design control points, each having target valuesof edging angle and inclination angle. The edging angle and inclinationangle at control point 1 are denoted by θ_(e) _(CP1) and θ_(i) _(CP1) ,respectively. Similarly, the edging angle and inclination angle atcontrol point 2 are denoted by θ_(e) _(CP2) and θ_(i) _(CP2) ,respectively. In general, the conditions at control point 2 should berepresentative of steeper edging and inclination angles than thosepresent at control point 1. As such, the value of θ_(e) _(CP2) should begreater than the value of θ_(e) _(CP1) , and the value of θ_(i) _(CP2)should be greater than the value of θ_(i) _(CP1) .

INPUTS_User_Params also includes the weight distribution between theleft and right feet of the athlete, which is defined as the fraction ofthe total athlete loading that is applied to the ski of interest. Underdesign point conditions, the fraction of the total athlete loading thatis applied to the ski of interest is denoted by w_(fDP). Similarly, thefractions of the total athlete loading that are applied to the ski ofinterest under the conditions of control points 1 and 2 are denoted byw_(fCP1) and w_(fCP2), respectively. The values of w_(fDP), w_(fCP1),and w_(fCP2) are typically set between 0.5 and 1.0, depending upon thetype of ski that is to be designed and the skiing technique that isemployed by the athlete. For example, a w_(fDP) value of 0.5 wouldindicate that the athlete is applying equal loads to the left hand andright hand skis; conversely, a w_(fDP) value of 1.0 would indicate thatthe athlete is applying all loads to the ski of interest, and the otherski is completely unloaded.

Although the camber formulation that is described herein may use twocontrol points, other aspects of the ski design system (such as theflexural stiffness profile) may only use a single design point at whichoptimization is to be carried out. For simplicity, the conditionspresent at control point 2 are set equal to those stipulated at thedesign point; hence, the value of θ_(e) _(CP2) is set equal to θ_(e)_(DP) , the value of θ_(i) _(CP2) is set equal to θ_(i) _(DP) , and thevalue of w_(fCP2) is set equal to w_(fDP). Conversely, the conditionspresent at control point 1 (θ_(e) _(CP1) , θ_(i) _(CP1) , and w_(fCP1))are explicitly stipulated by the user in INPUTS_User_Params.

INPUTS_User_Params also includes the angle of inclination of the piste(θ_(P)) upon which the athlete is skiing, wherein said angle is measuredbetween a horizontal plane and the surface of the snow. All of theaforementioned input angles (θ_(e) _(DP) , θ_(e) _(CP1) , θ_(e) _(CP2) ,θi_(DP), θ_(i) _(CP1) , θ_(i) _(CP2) , and θ_(P)) are provided aspositive values.

INPUTS_User_Params also includes the effective base length, nominalsidecut radius, taper, and minimum width of the ski at the waist.

INPUTS_User_Binding_Params includes the mounting position of the bindingsystem on the ski, the geometry of the ski, the stiffness of the bindinginterface plate system, and the binding release setting according toDeutsches Institut für Normung e.V. (DIN) standards.

INPUTS_Binding_Params includes user-stipulated parameters pertaining tothe boot-binding system that is used to fasten the athlete's feet to theski. These parameters include positions of the toe and heelanti-friction device (AFD) perches, binding geometry parameters, andparameters that pertain to the load-deflection responses of the variousmoving parts of the binding system (forward pressure system, toe-cupclamping force, heel-cup clamping force, brake pedal force, etc.). Inaddition, the user may provide values pertaining to the structuralcompliance (deflection per unit load) of the toe lug (C_Lug_f) and theheel lug (C_Lug_f) of the boot.

INPUTS_Engineering_Params includes the geometry of the shovel region,the geometry of the tail region, the minimum-permissible thickness ofthe core of the ski, the tip-rocker (slight reverse-camber within thevicinity of the tip of the ski, sometimes referred to as “early-rise”)geometry, and related optimization solution control options andparameters to afford the user with control over the characteristics ofthe optimization.

INPUTS_Snow_Model includes the parameters and/or coefficients thatpertain to a user-stipulated constitutive model of the snow, such as themodel presented by Federolf P., Roos M., Lüthi A., Dual J. (2010),Finite element simulation of the ski-snow interaction of an alpine skiin a carved turn, Sports Engineering, 12, 123-133, DOI10.1007/s12283-010-0038-z (“Federolf_et_al_2010”), which is herebyincorporated by reference. In some embodiments, the constitutive snowmodel includes plasticity during unloading, such as by employing animplementation similar to that presented by Federolf_et_al_2010.

INPUTS_Core_Materials includes the mechanical and physical properties ofeach material that is employed within the core of the ski.INPUTS_Core_Materials includes user-stipulated orthotopic materialmodels. Each of these orthotropic material models include the followingmechanical properties within the local 1-2-3 material coordinate system:elastic modulus parallel to the local 1-axis (E1); elastic modulusparallel to the local 2-axis (E2); elastic modulus parallel to the local3-axis (E3); shear modulus within the 1-2 plane (G12); shear moduluswithin the 2-3 plane (G23); shear modulus within the 1-3 plane (G13);Poisson's ratio from the 1-axis to the 2-axis (v12); Poisson's ratiofrom the 2-axis to the 3-axis (v23); and Poisson's ratio from the 1-axisto the 3-axis (v13). Each of the aforementioned orthotropic mechanicalproperties may be provided at two temperature control points, thusfacilitating linear interpolation of the orthotropic mechanicalproperties to the relevant conditions under which the ski is to beconstructed and optimized. In addition, each material model may includethe following physical properties: thermal strain along the local1-axis, as a function of temperature; thermal strain along the local2-axis, as a function of temperature; and thermal strain along the local3-axis, as a function of temperature. The aforementioned physicalproperties can be used to determine the effective thermal response ofeach material between any two temperature control points.

INPUTS_Core_Laminate includes the laminate architecture and the stackingsequence of the core of the ski. It is assumed that the core comprises avertical laminate, wherein the plane of each ply (1-2 plane of the localmaterial coordinate system) is parallel to the longitudinal axis of theski and is perpendicular to the base surface of the ski. The userdefines the material composition, thickness, and orientation (anglebetween the local 1-axis of the material and the longitudinal axis ofthe ski) of each ply within the laminate of the core.

INPUTS_Laminate_Materials includes the mechanical and physicalproperties of each material that is employed in the ski laminate.INPUTS_Laminate_Materials includes user-stipulated orthotropic materialmodels. Each of these orthotropic material models include the followingmechanical properties within the local 1-2-3 material coordinate system:elastic modulus parallel to the local 1-axis (E1); elastic modulusparallel to the local 2-axis (E2); shear modulus within the 1-2 plane(G12); shear modulus within the 2-3 plane (G23); shear modulus withinthe 1-3 plane (G13); and Poisson's ratio from the 1-axis to the 2-axis(v12). Each of the aforementioned orthotropic mechanical properties maybe provided at two temperature control points, thus facilitating linearinterpolation of the orthotropic mechanical properties to the relevantconditions under which the ski is to be constructed and optimized. Inaddition, each material model may include the following physicalproperties: thermal strain along the local 1-axis, as a function oftemperature; thermal strain along the local 2-axis, as a function oftemperature; and thermal strain along the local 3-axis, as a function oftemperature. The aforementioned physical properties can be used todetermine the effective thermal response of each material between anytwo temperature control points.

INPUTS_Laminate_Architecture includes the laminate architecture and thestacking sequence of the ski. It is assumed that the ski comprises ahorizontal laminate, wherein the plane of each ply (1-2 plane of thelocal material coordinate system) is approximately parallel to the basesurface of the ski. INPUTS_Laminate_Architecture includes materialcomposition, thickness, and orientation (angle between the local 1-axisof the material and the longitudinal axis of the ski) of each ply withinthe laminate of the ski, excluding the core material which is definedelsewhere.

INPUTS_Sublaminate_Plies is an optional set of inputs.INPUTS_Sublaminate_Plies enables users to define thin pre-curedlaminates that can be included within the overall laminate architectureof the ski. In essence, this optional input file allows the user topre-define small laminates that will then be amalgamated with the otherplies of the global laminate to form the total laminate of the ski.

INPUTS_Solid_Component_Materials includes the mechanical and physicalproperties of each material that is employed in the ski, other thanthose materials that are considered to be part of the laminate.INPUTS_Solid_Component_Materials includes user-stipulated orthotropicmaterial models. Each of these orthotropic material models include thefollowing mechanical properties within the local 1-2-3 materialcoordinate system: elastic modulus parallel to the local 1-axis (E1);elastic modulus parallel to the local 2-axis (E2); shear modulus withinthe 1-2 plane (G12); shear modulus within the 2-3 plane (G23); shearmodulus within the 1-3 plane (G13); and Poisson's ratio from the 1-axisto the 2-axis (v12). Each of the aforementioned orthotropic mechanicalproperties may be provided at two temperature control points, thusfacilitating linear interpolation of the orthotropic mechanicalproperties to the relevant conditions under which the ski is to beconstructed and optimized. In addition, each material model may includethe following physical properties: thermal strain along the local1-axis, as a function of temperature; thermal strain along the local2-axis, as a function of temperature; and thermal strain along the local3-axis, as a function of temperature. The aforementioned physicalproperties can be used to determine the effective thermal response ofeach material between any two temperature control points.

INPUTS_Solid_Component_Geometry includes the geometry and materialcomposition of solid components that are employed within theconstruction of the ski, such as: edges, edge anchors (teeth), andsidewalls.

INPUTS_Temperatures includes the user-stipulated cure temperature(TCure) of the ski, as well as three temperature control points: roomtemperature where the ski is to be built (TBuild), optimizationtemperature at which the performance of the ski is to be optimized(TOpt), and control temperature (TCont). The cure temperature (TCure) issimply set as the temperature at which the resin within the laminate ofthe ski is to be cured. The optimization temperature (TOpt) should beset as the temperature at which the user intends the ski to exhibitoptimum performance. The control temperature (TCont) should be set as anadditional temperature at which the ski will likely see service, but atwhich optimum performance is not to be expected (typically colder thanthe optimization temperature); this control temperature should generallybe set as the coldest temperature at which the ski is likely to be used.

CamRock_Offset is an optional set of inputs. The overall geometry of thecamber tooling (mould), when viewed in elevation, is referred to as the“camber mould elevation profile”, wherein said elevation profile existson a plane that is parallel to the longitudinal axis of the ski andperpendicular to the base surface of the ski. The “camber mouldcurvature profile” defines the curvature of the camber mould elevationprofile at any position along the length of the ski, wherein saidcurvature is measured within a plane that is parallel to thelongitudinal axis of the ski and perpendicular to the base surface ofthe ski. CamRock_Offset is an additional offsetting curvature profilethat is added to the analytically determined camber mould curvatureprofile. For example, CamRock_Offset may be employed if initialmanufacturing runs empirically illustrate a need for a modification tothe camber mould elevation profile in order to achieve the targetedoverall camber and rocker elevation profile of the ski. The need forsuch an offsetting curvature profile may be caused by residual thermalstresses in some of the constituent materials in the ski, or by othercomplex sources of processing induced stresses that are not fullycaptured by the assumptions that are made by system 2000.

Sidecut_Offset is an optional set of inputs. By default, system 2000will automatically generate a baseline sidecut profile that willtheoretically yield a linear snow penetration depth profile. TheSidecut_Offset input file contains a user-stipulated array comprisingordered pairs of tangential positions and sidecut offsetting distances;this array serves to modify the sidecut profile from the baselinesidecut profile that is initially assumed by system 2000.

At 2010A, ski design system 2000 calculates the athlete loading on theski. That is, the ski design system 2000 calculates the component of thetotal loading that acts perpendicular to the base surface of the ski.For example, the ski design system 2000 may calculate the total loadingas follows:

A total load factor is defined as the multiple of the athlete's staticself-weight that is imposed upon the skis due to the vector sum of theathlete's weight and the centripetal acceleration, assuming that theathlete is balanced at his/her design point inclination angle (θ_(i)_(DP) ) under steady-state turning conditions. Under design pointconditions, the total load factor is calculated as:

${a_{LFtotDP} = \frac{\cos \left( \theta_{P} \right)}{\cos \left( \theta_{i_{DP}} \right)}},$

where θ_(P) is the angle of inclination of the piste (measured between ahorizontal plane and the surface of the snow), and θ_(i) _(DP) is theathlete's inclination angle at the design point. Under design pointconditions, the component of the total load factor that actsperpendicular to the base surface of the skis is then calculated as:a_(LFnDP)=a_(LFtotDP) cos(θ_(e) _(DP) −θ_(i) _(DP) ), where θ_(e) _(DP)is the edging angle of the ski at the design point.

Combining the aforementioned equations, the component of the total loadfactor that acts perpendicular to the base surface of the skis isdirectly calculated as follows:

$a_{LFnDP} = {\frac{{\cos \left( \theta_{P} \right)}{\cos \left( {\theta_{e_{DP}} - \theta_{i_{DP}}} \right)}}{\cos \left( \theta_{i_{DP}} \right)}.}$

Similarly, under the conditions of control point 1, the component of thetotal load factor that acts perpendicular to the base surface of theskis can be calculated as follows:

${a_{{LFnCP}\; 1} = \frac{{\cos \left( \theta_{P} \right)}{\cos \left( {\theta_{e_{{CP}\; 1}} - \theta_{i_{{CP}\; 1}}} \right)}}{\cos \left( \theta_{i_{{CP}\; 1}} \right)}},$

where θ_(e) _(CP1) is the edging angle of the ski at control point 1,and θ_(i) _(CP1) is the athlete's inclination angle at control point 1.Finally, under the conditions of control point 2, the component of thetotal load factor that acts perpendicular to the base surface of the skican be calculated as follows:

${a_{{LFnCP}\; 2} = \frac{{\cos \left( \theta_{P} \right)}{\cos \left( {\theta_{e_{{CP}\; 2}} - \theta_{i_{{CP}\; 2}}} \right)}}{\cos \left( \theta_{i_{{CP}\; 2}} \right)}},$

where θ_(e) _(CP2) is the edging angle of the ski at control point 2,and θ_(i) _(CP2) is the athlete's inclination angle at control point 2.

Then, under design point conditions, the component of the total forcethat acts perpendicular to the base surface of the ski is calculated as:F_(nDP)=w_(fDP)*m*g*a_(LFnDP), where m is the mass of the athlete, g isacceleration due to gravity (approximately 9.807 newtons per kilogram),and w_(fDP) is the fraction of the total athlete loading that is appliedto the ski of interest under design point conditions. Similarly, underthe conditions of control point 1, the component of the total force thatacts perpendicular to the base surface of the ski is calculated as:F_(nCP1)=w_(fCP1)*m*g*a_(LFnCP1), where w_(fCP1) is the fraction of thetotal athlete loading that is applied to the ski of interest under theconditions of control point 1. Finally, under the conditions of controlpoint 2, the component of the total force that acts perpendicular to thebase surface of the ski is calculated as:F_(nCP2)=w_(fCP2)*m*g*a_(LFnCP2), where w_(fCP2) is the fraction of thetotal athlete loading that is applied to the ski of interest under theconditions of control point 2.

In some embodiments, the user may directly provide values of F_(nDP),F_(nCP1), and/or F_(nCP2) within the INPUTS_User_Params input file; ifthe user chooses to do so, then these values can simply override anycalculated values of F_(nDP), F_(nCP1), and/or F_(nCP2).

At 2010B, the ski design system 2000 discretizes the ski. The ski can bediscretized into a finite number of segments over the length of the ski.In some embodiments, at least 300 segments are used for thisdiscretization.

For example, each segment can be represented by a unidimensional beamelement that is bound by a pair of nodes (one node at the forwardextremity of the element, and one node at the aft extremity of theelement). Each beam element and/or node can be assigned a variety ofcharacteristics including, but not limited to: vector position andorientation in space, evaluated and stored at the elevation of the baseof the ski and at the elevation of the flexural neutral axis of the ski;tangential position (Xmid) along the length of the ski (having an originat the mid-base of the ski), evaluated and stored at the elevation ofthe base of the ski (XmidB) and at the elevation of the flexural neutralaxis of the ski (XmidNA); local width of the ski; flexural (bending)stiffness; torsional (twisting) stiffness; camber curvature, evaluatedand stored at the elevation of the base of the ski and at the elevationof the flexural neutral axis of the ski; total curvature at the designpoint, evaluated and stored at the elevation of the base of the ski andat the elevation of the flexural neutral axis of the ski; local relativeangle of twist at the design point, initially set to zero and thenrevised later; and local edging angle (thetaEloc_rel) of the ski.

The example discretization process creates a series of arrays that canbe used to store the data pertaining to the various characteristics ofeach beam element and/or node along the length of the ski. As will beshown in the subsequent descriptions, some of these characteristicarrays can be assigned initial values during the initialization phase ofthe solution (such as tangential position of each node, width profile,camber curvature profile, and total curvature profile at the designpoint), but these characteristics may be subject to change as theoptimization loop converges upon a design solution. Conversely, some ofthe characteristic arrays may not be assigned any initial values (suchas flexural stiffness, torsional stiffness, and angle of twist), and maysimply be given values of zero until the first iteration of theoptimization loop has generated enough data to begin populating thesearrays.

At 2010C, the ski design system 2000 initializes the local edging angleprofile over the length of the ski at the design point. The ski designsystem 2000 can assume that there is zero torsional deformation at thedesign point.

For example, an array (thetaEloc_rel_init) can be created in order tostore the local edging angle at each node along the length of the ski,while ignoring the effects of any torsional deformations. In order toensure that the local b orientation vectors point toward the surface ofthe snow (thus simplifying subsequent calculations), it can be assumedthat the ski is optimized in the context of a right hand turn. In theinterest of observing right hand rule conventions, each entry in thethetaEloc_rel_init array can be given a negative value having amagnitude that is equal to that of the global edging angle of the ski atthe design point (θ_(e) _(DP) ).

Continuing with the above example, another array (thetaEloc_rel) canthen be defined as that which will store the local edging angle at eachnode along the length of the ski, while accounting for the effects oftorsional deformations. Initially, it is assumed that the ski exhibitszero torsional deformation under design point conditions; as such, thethetaEloc_rel array is initially set equal to the thetaEloc_rel_initarray. As the optimization loop of method 2020 converges upon a designsolution, the thetaEloc_rel array can be adjusted in order to reflectincreasingly accurate assessments of the torsional deformation of theski. Conversely, the thetaEloc_rel_init array may not be altered fromits initial definition.

At 2010D, the ski design system 2000 generates an initial assumption ofthe total deformed shape of the ski at the design point. Initially, itcan be assumed that the ski exhibits zero torsional deformation underdesign point conditions.

For example, an initial assumed total curvature profile of the ski atthe design point can be defined at each X coordinate within theeffective base region of the ski (with an assumed origin at the mid-baseof the ski), as follows:

${\varnothing_{{TotDP}_{init}} = {\frac{R_{SC}^{2}{\cos^{2}\left( \theta_{e_{DP}} \right)}{\sin \left( \theta_{e_{DP}} \right)}}{\left\lbrack {{R_{SC}^{2}{\cos^{2}\left( \theta_{e_{DP}} \right)}} - X^{2}} \right\rbrack^{\frac{3}{2}}}\left\lbrack {1 + \frac{X^{2}{\sin^{2}\left( \theta_{e_{DP}} \right)}}{{R_{SC}^{2}{\cos^{2}\left( \theta_{e_{DP}} \right)}} - X^{2}}} \right\rbrack}^{- \frac{3}{2}}},$

where θ_(e) _(DP) is the edging angle of the ski at the design point(taken as a positive value), and R_(SC) is the user-stipulated nominalsidecut radius of the ski. At the mid-base of the ski (where X=0), thecurvature of the aforementioned assumed deformed shape simplifies to thefollowing expression:

$\varnothing_{{TotDP}_{{{init}\; x} = 0}} = {\frac{\tan \left( \theta_{e_{DP}} \right)}{R_{SC}}.}$

The aforementioned equation for Ø_(TotDP) _(init) is based upon theassumption that a ski having a constant sidecut radius of R_(SC) wouldbend into a partial ellipse (elliptical arc) geometry when oriented atan edging angle of θ_(e) _(DP) and de-cambered onto a flat surface; theexpression for Ø_(TotDP) _(init) is then derived as the curvatureprofile of said partial ellipse. Since it has been assumed that thedeformed shape of the ski is that of a partial ellipse geometry, it ispossible to express this partial ellipse geometry with reference to alocal two-dimensional Cartesian coordinate system that lies within aplane that is oriented perpendicular to the base surface of the ski,wherein a local x-axis is oriented parallel to the global X-axis of theski, a local z_(TotDP) _(init) -axis is oriented perpendicular to thelocal x-axis, and the origin of the coordinate system is positioned onthe base surface of the ski at the mid-base position. Recognizing that xand X are equivalent, the geometry of the assumed partial ellipsedeformed shape can then be expressed within this local two-dimensionalcoordinate system, as follows: z_(TotDP) _(init) =[R_(SC) cos(θ_(e)_(DP) )−(R_(SC) ² cos²(θ_(e) _(DP) )−X²)^(1/2)]sin(θ_(e) _(DP) ).

While the aforementioned expressions for Ø_(TotDP) _(init) and z_(TotDP)_(init) are provided in terms of the X coordinate along the length ofthe ski, it may be more convenient to carry out subsequent calculationsin terms of the tangential position (XmidB) along the curvilinear lengthof the base of the ski. As such, it may be necessary to find the segmentlength of each element within the assumed partial ellipse geometry,numerically integrate these segment lengths over the length of thepartial ellipse geometry to find the tangential position thatcorresponds to each X coordinate, and then interpolate the tangentialcoordinates (XmidB) of the ski along the partial ellipse geometry suchthat values of Ø_(TotDP) _(init) and z_(TotDP) _(init) can be assignedto each corresponding tangential coordinate (XmidB) along the length ofthe base of the ski.

The aforementioned formulation for Ø_(TotDP) _(init) is employed becauseit theoretically results in a snow trace (the geometry of the effectiveline of contact between the ski and the snow) that exhibits a constantradius (round) geometry. In general, a carved turn is achieved when theforebody of a ski (segment of the ski that is ahead of the mid-basepoint) cuts a groove (carved groove) in the snow, and the afterbody ofthe ski (segment of the ski that is behind the mid-base point) deformsto an arced shape such that it can smoothly pass through the carvedgroove that was created by the forebody of the ski. The radius of theresulting carved turn is equal to the radius of the path that is tracedby this carved groove, which is approximately dictated by the averageradius of curvature of the snow trace within afterbody region of theski. As such, in order to carve a smooth turn with minimum drag, theafterbody may adopt a deformed shape that yields a constant radius snowtrace geometry. Conversely, the forebody of the ski may adopt a deformedshape having a somewhat smaller radius of curvature in order to keep thetip of the ski above the surface of the snow as the carved groove isbeing cut. For this reason, the snow trace geometry that is exhibitedwithin the forebody of the ski may differ slightly from the geometry ofthe carved groove.

Tip-rocker (slight reverse-camber within the vicinity of the tip of theski, sometimes referred to as “early-rise”) may be necessary in order tocut the trench (carved groove) through which the aft regions of the skiwill subsequently pass, while simultaneously allowing the aft regions ofthe ski to adopt a deformed shape that is conducive for achieving asmooth and efficient carved turn. As such, the height of this tip-rockercan be defined as a function of the depth of the carved groove that isto be cut by the ski. In soft snow conditions, a deep carved groove willbe cut, thus necessitating a relatively large amount of tip-rocker inorder to effectively cut the carved groove without generating undue dragon the shovel of the ski. Conversely, in firm or icy snow conditions,very little tip-rocker is needed because the carved groove will berelatively shallow.

With the aforementioned considerations in mind, professional downhillski racers may be well advised to bring numerous skis to each race, eachwith different amounts of tip-rocker. Prior to the race, a series ofsimple snow-hardness measurements can be taken, and the results of thesesnow hardness measurements can be used to determine how much tip-rockeris needed, and in turn, which pair of skis should be used for thatparticular race. The consequences of selecting a non-optimal amount oftip-rocker could be significant. If a pair of race skis having excessivetip-rocker is selected (snow is too firm for the amount of tip-rockerexhibited by the skis), then the athlete will likely find that he/shehas insufficient edge engagement in the firm or icy snow conditions,thus resulting in poor grip at steep edging angles. Conversely, if apair of race skis having insufficient tip-rocker is selected (snow istoo soft for the amount of tip-rocker exhibited by the skis), then theskis will not adopt a smooth deformed shape in the snow, and will tendto plow a lot of snow in their shovel regions, thus generating unduedrag and lost speed during carved turns.

As described above, the ski design system 2000 may allow the user tostipulate a tip-rocker length (L_(ER)) and a maximum tip-rockercurvature (Ø_(MaxER)) These values can be used to generate a tip-rockercurvature profile (Ø_(TipRock)) that begins with the value of Ø_(MaxER)at the forward-most point within the effective base region of the ski,and linearly reduces to a curvature of zero at a position that ispositioned a distance of L_(ER) aft of the forward-most point within theeffective base region of the ski, thus yielding an Euler spiral(clothoid) tip-rocker geometry. The value of Ø_(MaxER) can also be setto zero at all positions that are positioned a distance greater thanL_(ER) aft of the forward-most point within the effective base region ofthe ski. While some embodiments assume an Euler spiral tip-rockergeometry, other embodiments may utilize alternative tip-rockergeometries (such as a constant radius arc).

Once this tip-rocker geometry is defined, the initial assumed totalcurvature profile of the ski at the design point can be modified asfollows: Ø_(TotDP)=Ø_(TotDP) _(init) +Ø_(TipRock). It should be notedthat the user-stipulated tip-rocker (Ø_(TipRock)) is not employed in amanner that directly modifies the camber elevation profile of the ski;on the contrary, the user-stipulated tip-rocker (Ø_(TipRock)) isemployed to adjust the assumed total curvature profile of the ski at thedesign point, and then the ski design system 2000 is then tasked withdetermining how the ski must be designed in order to achieve said totalcurvature profile under design point conditions.

The aforementioned total curvature profile may be initially assumed tohave been created in the absence of the ski exhibiting any torsionaldeformations. This initial assumed total curvature profile along thebase of the ski (Ø_(TotDP)), which neglects torsional deformations, willherein be referred to as PhiTotalB_NoTwist, and can be stored withoutchange. Conversely, the true total curvature profile along the length ofthe base of the ski (PhiTotalB) can initially be set equal toPhiTotalB_NoTwist, but may be subject to change as the ski design system2000 refines the design of the ski and properly accounts for the effectsof torsional deformation (thetaEloc_rel). In essence, the assumedPhiTotalB_NoTwist curvature profile, in conjunction with the initialassumption of zero torsional deformation, embodies one of theoptimization conditions of ski design system 2000, wherein the optimizedski design is intended to yield a snow trace geometry that is of anapproximately constant radius arc under design point conditions.Notwithstanding the foregoing, the introduction of tip-rocker wouldcause the snow trace geometry to deviate from that of a constant radiusarc, and this may be beneficial for reasons discussed elsewhere herein.

It should be noted that the aforementioned formulation for the totalcurvature profile may only be created within the effective base lengthof the ski. As such, additional nodes and elements can then be added torepresent the shovel and tail regions of the ski, and these regions canbe assigned user-stipulated curvatures as necessary to create the upwardcurving geometries of these features. The nodes that are added torepresent the shovel and tail regions of the ski will subsequently bereferred to as “shovel nodes” and “tail nodes”, respectively. Similarly,the elements that are added to represent the shovel and tail regions ofthe ski will subsequently be referred to as “shovel elements” and “tailelements”, respectively.

At 2010E, the ski design system 2000 establishes an initial assumedcamber elevation profile, which is represented by a camber curvatureprofile.

For example, the initial camber curvature profile (ignoring early-riseand/or rocker) can be determined by invoking a function called“CamProfile” that generates a camber elevation profile based upon theeffective base length of the ski, the nominal sidecut radius of the ski,the edging angle at the design point (control point 2), the edging angleat control point 1, the component of the total force that acts normal tothe base surface of the ski at the design point (control point 2), andthe component of the total force that acts normal to the base surface ofthe ski at control point 1. The goal of the CamProfile function is toset the camber elevation profile such that a flexural stiffness profilethat would yield an ideal deformed shape (perhaps corresponding to anapproximately constant radius snow trace profile) at the design point(control point 2) would also yield a nearly ideal deformed shape(perhaps corresponding to an approximately constant radius snow traceprofile) at control point 1. This can be achieved by means of thefollowing procedure.

The total curvature at the mid-base of the ski can be evaluated atcontrol point 1 by employing an equation of a similar form to that usedfor Ø_(TotDP) _(init x=) 0, as follows:

${\varnothing_{{TotCP}\; 1_{x = 0}} = \frac{\tan \left( \theta_{e_{{CP}\; 1}} \right)}{R_{SC}}},$

where θ_(e) _(CP1) is the edging angle at control point 1 (taken as apositive value), and R_(SC) is the user-stipulated nominal sidecutradius of the ski. Similarly, the total curvature at the mid-base of theski can be evaluated at control point 2, as follows:

${\varnothing_{{TotCP}\; 2_{x = 0}} = \frac{\tan \left( \theta_{e_{{CP}\; 2}} \right)}{R_{SC}}},$

where θ_(e) _(CP2) is the edging angle at control point 2 (taken as apositive value).

If it is assumed that all snow loading (w) is uniformly distributedalong the effective base length of the ski at both control points, thenit can be assumed that a linear relationship exists between the totalcurvature at the mid-base of the ski and the total force that actsnormal to the base surface of the ski; the slope of this line can beexpressed as follows:

${m_{{{CP}\; 1} - 2} = \frac{\varnothing_{{TotCP}\; 2_{x = 0}} - \varnothing_{{TotCP}\; 1_{x = 0}}}{F_{{nCP}\; 2} - F_{{nCP}\; 1}}},$

where F_(nCP1) is the component of the total force that acts normal tothe base surface of the ski at control point 1, and F_(nCP2) is thecomponent of the total force that acts normal to the base surface of theski at control point 2. Combining the aforementioned equations, theexpression for m_(CP1-2) can be rewritten as follows:

$m_{{{CP}\; 1} - 2} = {\frac{{\tan \left( \theta_{e_{{CP}\; 2}} \right)} - {\tan \left( \theta_{e_{{CP}\; 1}} \right)}}{R_{SC}\left\lbrack {F_{{nCP}\; 2} - F_{{nCP}\; 1}} \right\rbrack}.}$

Recalling that m_(CP1-2) represents the slope of a linear relationshipbetween the total curvature at the mid-base of the ski and the totalforce that acts normal to the base surface of the ski, the mid-basecurvature that is dictated by this function when the total force thatacts normal to the base surface of the ski is equal to zero can becalculated as follows: b_(CP1-2)=Ø_(TotCP1) _(x=) 0−m_(CP1-2)*F_(nCP1).The goal is to find a camber elevation profile that will ensure that thelinear load versus curvature response of the ski passes through theorigin as well as both of the two selected control points. As such, theselected camber elevation profile should exhibit a mid-base curvaturethat is equal to b_(CP1-2).

Substituting the relevant equations into the aforementioned formula forb_(CP1-2), the camber curvature at the mid-base of the ski (where X=0)can be found as follows:

$\varnothing_{{Cam}_{x = 0}} = {\frac{1}{R_{SC}}{\left( {{\tan \left( \theta_{e_{{CP}\; 1}} \right)} - \frac{\left\lbrack {{\tan \left( \theta_{e_{{CP}\; 2}} \right)} - {\tan \left( \theta_{e_{{CP}\; 1}} \right)}} \right\rbrack F_{{nCP}\; 1}}{F_{{nCP}\; 2} - F_{{nCP}\; 1}}} \right).}}$

Recalling that control point 2 is equal to the design point, the valueof F_(nCP2) can be set equal to F_(nDP). By assuming that the camberelevation profile is of the same form as that assumed in the expressionfor Ø_(TotDP) _(init) ′ the curvature profile of the camber elevationprofile at any X coordinate along the length of the effective baseregion of the ski can then be calculated as:

${\varnothing_{cam} = {S_{{Cam}\; 0}{\frac{R_{SC}^{2}{\cos^{2}\left( \theta_{Cam} \right)}{\sin \left( \theta_{Cam} \right)}}{\left\lbrack {{R_{SC}^{2}{\cos^{2}\left( \theta_{Cam} \right)}} - X^{2}} \right\rbrack^{\frac{3}{2}}}\left\lbrack {1 + \frac{X^{2}{\sin^{2}\left( \theta_{Cam} \right)}}{{R_{SC}^{2}{\cos^{2}\left( \theta_{Cam} \right)}} - X^{2}}} \right\rbrack}^{- \frac{3}{2}}}},$

where S_(Cam0) is taken as +1 if Ø_(Cam) _(x=0) is positive and S_(Cam0)is taken as −1 if Ø_(Cam) _(x=0) is negative, and where θ_(Cam) is anon-physical phantom edging angle that is calculated using arearrangement of a similar equation to that used for Ø_(TotDP)_(init x=0) , as follows: θ_(Cam)=S_(Cam0) tan⁻¹(R_(SC)Ø_(Cam) _(x=0) ).

In most cases, the values of Ø_(Cam) _(x=0) and S_(Cam0) will both benegative. While the aforementioned expression for Ø_(Cam) is provided interms of the X coordinate along the length of the ski, it may be moreconvenient to carry out subsequent calculations in terms of thetangential position (XmidBcam) along the curvilinear length of the basesurface of the ski under camber conditions. As such, it may be necessaryto find the segment length of each element within the assumed partialellipse camber elevation profile, numerically integrate these segmentlengths over the length of the partial ellipse geometry to find thetangential position that corresponds to each X coordinate, and theninterpolate the tangential coordinates (XmidBcam) of the ski along thepartial ellipse geometry such that a camber curvature value (Ø_(Cam))can be assigned to each corresponding tangential coordinate (XmidBcam)along the length of the base of the ski.

It should be noted that the aforementioned camber curvature profile mayonly be created within the effective base region of the ski. As such,the shovel nodes and tail nodes can be assigned user-stipulatedcurvature values as necessary to create the upward curving geometries ofthese features. Ultimately, two camber geometries can be found: Cam0represents the camber elevation profile of the unloaded ski withoutboots installed in the binding, whereas Cam1 represents the camberelevation profile of the unloaded ski with boots installed in thebindings and accounting for the loads that the boot-binding systemimposes upon the ski. Initially, these camber geometries are assumed tobe equal, but may be updated later as the ski design system 2000converges upon an optimized design solution that accounts for theloading that the boot-binding system imposes upon the ski. The user maybe given the option to stipulate whether the targeted camber elevationprofile is to be achieved with or without boots installed in thebindings. Hence, the user may decide whether the targeted camberelevation profile is to be represented by Cam0 or Cam1.

At 2010F, the ski design system 2000 then initializes the width profileof the ski.

For example, the width profile can be initialized as a function of theassumed deformed shape of the ski at the design point, as well as theuser stipulated values of effective base length, taper, and waist width.In this example embodiment, it is assumed that the ski is symmetricabout its longitudinal axis; hence, the ski has the same sidecut profilealong both of its edges. The ski design system 2000 can calculate areversed nominal sidecut depth at each tangential coordinate (XmidB)along the length of the base of the ski, as follows:

$d_{SC\_ rev} = {\frac{Z_{{TotDP}_{init}}}{Z\; \left( \theta_{e_{DP}} \right)}.}$

An initial unmodified sidecut profile can then be calculated as follows:d_(SC_init)=d_(SC_rev_Max)−d_(SC_rev), where d_(SC_rev_Max) is themaximum value within the d_(SC_rev) array.

The d_(SC_init) array represents a sidecut profile that wouldtheoretically result in a uniform (constant) snow penetration depthprofile, provided that said ski is loaded at its mid-base point andexhibits the total deformed shape that has been assumed at the designpoint (said assumed deformed shape corresponding to Ø_(TotDP) _(init) ),wherein said assumed deformed shape does not include any torsionaldeformation. If said ski were to be loaded at a point that is notaligned with the longitudinal position of its mid-base point, then thesnow penetration depth profile would remain quasi-linear, but wouldbecome inclined such that it adopts a quasi-constant slope over thelength of the ski, thus ensuring that the centroid of the snow loadingdistribution remains coincident with the load application point.

In some cases, it may be desirable to design a ski that exhibits anon-linear snow penetration depth profile at the design point, whilestill achieving a snow trace (the effective line of contact between theski and the snow) that exhibits a desired geometry (perhaps anapproximately constant radius arc). For example, the user may wish todesign a ski that exhibits a W-shaped edge loading distribution at thedesign point, wherein the edge loading distribution exhibits localmaxima values in the vicinities of the shovel, the tail, and themid-base regions of the ski. As such, the user may provide an optionalinput file (Sidecut_Offset) that contains ordered pairs of tangentialpositions and corresponding sidecut offset distances. Upon reading theSidecut_Offset input file, the ski design system 2000 can then employ acubic interpolation function to find an array (d_(SC_offset)) thatcontains the sidecut offset value that corresponds to each tangentialcoordinate (XmidB) along the length of the ski. A modified sidecutprofile is then formulated as follows:d_(SC_mod)=d_(SC_init)+d_(SC_offset).

If the user does not provide a Sidecut_Offset input file, then it isassumed that d_(SC_offset) is a zero-vector, and the d_(SC_mod) array issimply set equal to the d_(SC_init) array. The sidecut profile can thenbe adjusted such that the minimum sidecut value is equal to zero, asfollows: d_(SC_0)=d_(SC_mod)−d_(SC_mod_Min), where d_(SC_mod_Min) is theminimum value within the d_(SC_mod) array. A taper angle is defined asfollows:

${\theta_{taper} = {\tan^{- 1}\left( \frac{T}{2\; L_{b}} \right)}},$

where T is the taper of the ski, and L_(b) is the effective base lengthof the ski. The d_(SC_mod) sidecut profile is then rotated about theglobal Z-axis of the ski, by an angle of θ_(taper), and a cubicinterpolation function is employed to find an array (d_(SC_T)) thatcontains the rotated (tapered) sidecut profile values that correspond toeach tangential coordinate (XmidB) along the length of the ski.

Finally, the overall width profile of the ski can be calculated asfollows: B_(XmidB)=B_(W)+2 (d_(SC_T_Max)−d_(SC_T)), where B_(XmidB) isthe width value at each tangential coordinate (XmidB) along the lengthof the effective base region of the ski, B_(W) is the user-stipulatedwaist width of the ski, and d_(SC_T_Max) is the maximum value within thed_(SC_T) array.

It should be noted that the aforementioned procedure may only generate awidth profile within the effective base region of the ski. As such, afunction called “ShovelTailWidths” can be invoked in order to assignlocal width profiles to the shovel nodes and tail nodes such that theplanform geometries of these regions satisfy the geometric relationshipsthat were stipulated by the user.

At 2010G, the ski design system 2000 calculates mechanical and physicalproperties of each constituent material in the ski at each of thetemperature control points.

For example, at each temperature control point, the ski design system2000 can employ a linear interpolation function to find the mechanicalproperties of each material at the relevant temperature control point.The ski design system 2000 can calculate the change in temperature fromthe cure temperature to the relevant temperature control point, andcalculate a secant Coefficient of Linear Thermal Expansion (“CLTE”)between the cure temperature and the relevant temperature control point.This calculation may use a detailed thermal strain versus temperatureresponse that envelops the relevant temperature range.

For example, suppose the user provides the elastic modulus of a materialin the 1 direction at a temperature of 295 kelvins (E1 ₂₉₅) and at atemperature 253 kelvins (E1 ₂₅₃), and provides the thermal strain versustemperature response of that same material between 333 kelvins and 250kelvins. Suppose also that the user has stipulated that the ski is to becured at a temperature of 320 kelvins, and is to be optimized for use ata temperature of 265 kelvins. The elastic modulus of the material in the1 direction at the optimization temperature of 265 kelvins is thenapproximated as:

${E\; 1_{265}} = {{E\; 1_{253}} + {\left( {{E\; 1_{295}} - {E\; 1_{253}}} \right){\frac{\left( {265 - 253} \right)}{\left( {295 - 253} \right)}.}}}$

The secant CLTE that is to be used between the cure temperature and theoptimization temperature is calculated as:

${\alpha_{265 - 320} = \frac{\left( {ɛ_{320} - ɛ_{265}} \right)}{\left( {320 - 265} \right)}},$

where ε₃₂₀ is the thermal strain taken from the thermal strain versustemperature response at the cure temperature (320 kelvins), and ε₂₆₅ isthe thermal strain taken from the thermal strain versus temperatureresponse at the optimization temperature (265 kelvins).

Because of the nonlinear nature of most thermal strain versustemperature responses that cover a broad temperature range, the use ofthis secant CLTE value may be restricted to calculating thermalexpansions between the two specific temperatures that were used for thecalculation of the secant CLTE (in this case, between 320 kelvins and265 kelvins). For example, use of the aforementioned α₂₆₅₋₃₂₀ CLTE valueto find the thermal expansion caused by a temperature change from 320kelvins to 280 kelvins could yield erroneous results.

By implementing a collection of secant CLTE values for each materialwithin the laminate of the ski, the ski design system 2000 is able toemploy Classical Laminated Plate Theory (see Jones R. M., (1975),Mechanics of composite materials, Hemisphere Publishing Corporation, NewYork, N.Y., USA (“Jones_1975”), and/or Mallick P. K., (2008),Fiber-Reinforced Composites: Materials, Manufacturing, and Design, CRCPress—Taylor and Francis Group, Boca Raton, Fla., USA, 3rd edition, ISBN978-0-8493-4205-9 (“Mallick_2008”), each of which is hereby incorporatedby reference) for the calculation of thermal strains and thermalcurvatures of the full ski laminate, irrespective of the possibilitythat one or more of the materials in the laminate may exhibit anonlinear thermal strain versus temperature response.

At 2010H, the ski design system 2000 establishes a relationship betweencore thickness, ski width, and flexural stiffness at the optimizationtemperature control point.

For example, the ski design system 2000 can loop over a range ofpossible ski widths and core thicknesses; each combination of ski widthand core thickness can be processed and evaluated by a function called“EI_JG” in order to determine the flexural and torsional stiffnessesthat would result from the given combination of ski width and corethickness, in light of the stipulated material composition and laminatearchitecture of the ski. Once the E1 JG function has assessed theflexural and torsional stiffness values that correspond to eachcombination of ski width and core thickness, a series of response curvescan be generated such that the ski design system 2000 can laterinterpolate between these curves in order to determine what corethickness would be required in order to achieve a targeted flexuralstiffness for a given ski width. FIG. 11 shows an example plot 5000illustrating examples of these response curves.

The EI_JG function can be invoked for the purpose of determining thelocal flexural stiffness and local torsional stiffness that would resultfrom a given combination of ski width and core thickness, in light ofthe stipulated material composition and laminate architecture of theski. The EI_JG function performs laminate stiffness calculations inaccordance with Classical Laminated Plate Theory, which is described inmany reference books, including Jones R. M., (1975), Mechanics ofcomposite materials, Hemisphere Publishing Corporation, New York, N.Y.,USA (“Jones_1975”), and Mallick P. K., (2008), Fiber-ReinforcedComposites: Materials, Manufacturing, and Design, CRC Press—Taylor andFrancis Group, Boca Raton, Fla., USA, 3rd edition, ISBN978-0-8493-4205-9 (“Mallick_2008”), each of which are herebyincorporated by reference.

The EI_JG function adds the effects of solid components that are notpart of the laminate (such as edges and sidewalls) by employing themethod presented by Clifton P., (2011), Investigation and customisationof snowboard performance characteristics for different riding styles,Doctor of Philosophy (PhD), Aerospace, Mechanical and ManufacturingEngineering, RMIT University (“Clifton_2011”) and Brennan S., (2003),Modeling the mechanical characteristics and on-snow performance ofsnowboards, Doctor of Philosophy (PhD), Department of Aeronautics andAstronautics, Stanford University. (“Brennan_2003”), each of which ishereby incorporated by reference.

The EI_JG function modifies torsional stiffness using the modelpresented by Honickman H., Johrendt J., and Frise P., (2014), On thetorsional stiffness of thick laminated plates, Journal of CompositeMaterials, 48(21): 2639-2655, DOI: 10.1177/0021998313501919(“Honickman_et_al_2014”), which is hereby incorporated by reference.

In addition, the EI_JG function also carries out a secondary calculationof torsional stiffness by treating the ski as a closed thin-walled tube(torsion-box) comprising a lower skin (base and lower reinforcingplies), an upper skin (topsheet and upper reinforcing plies), and twosidewalls, and assuming that torsional stiffness is achieved as a resultof the generation of shear flow at the median-line (mid-plane) of eachof these thin shell elements, as described in many reference books,including Budynas R. G., (1999), Advanced Strength and Applied StressAnalysis, McGraw-Hill, The McGraw-Hill Companies, New York, N.Y., USA,2^(nd) Edition, ISBN 978-0-07-008985-3 (“Budynas_1999”), which is herebyincorporated by reference.

The EI_JG function then compares the two aforementioned torsionalstiffness values (that which was calculated in accordance withHonickman_et_al_2014 and that which was calculated in accordance withclosed thin-walled tube theory), and the greater of those two values isstored as that which will serve to represent the local torsionalstiffness of the ski.

It is important to note that the aforementioned methodology that isemployed by the EI_JG function is not intended to be restrictive. Thepurpose of the EI_JG function is to determine the local flexuralstiffness and local torsional stiffness that would result from a givencombination of ski width and core thickness, in light of the stipulatedmaterial composition and laminate architecture of the ski; however, someembodiments may employ a multitude of alternative analytical and/orcomputational methods in order to achieve that same purpose. In fact,some of such alternative analytical and/or computational methods mayyield more accurate results than the aforementioned methods employedherein. For example, some alternative embodiments of the EI_JG functioncould employ the Variational Asymptotic Method (VAM) to calculate thelocal section constants (such as flexural stiffness and torsionalstiffness), as described in numerous publications, such as: Cesnik C,Hodges D, and Sutyrin V, (1996) Cross-sectional analysis of compositebeams including large initial twist and curvature effects, AIAA Journal,34(9): 1913-1920, DOI: 10.2514/3.13325 (“Cesnik_et_al_1996”), Cesnik Cand Hodges D, (1997) VABS: A New Concept for Composite Rotor BladeCross-Sectional Modeling, Journal of the American Helicopter Society,42(1): 27-38, DOI: 10.4050/JAHS.42.27 (“Cesnik_and_Hodges_1997”), and YuW, Volovoi V, Hodges D, and Hong X, (2002) Validation of the variationalasymptotic beam sectional analysis (VABS), AIAA Journal, 40(10):2105-2113, DOI: 10.2514/2.1545 (“Yu_et_al_2002”), each of which ishereby incorporated by reference.

At 2010I, the ski design system 2000 initializes the assumed totalcurvature profile at the flexural neutral axis (PhiTotalNA), and thevarious camber curvature profiles at the flexural neutral axis(PhiCamRock0NA, PhiCam0NA, PhiCamRock1NA, PhiCam0NA). For example, theski design system 200 can invoke a function called “PhiZOffset”.

At 2010I, the ski design system 2000 also initializes the assumedtangential coordinate system at the flexural neutral axis (XmidNA). Forexample, the ski design system can invoke a function called“XmidZOffset”. The elevation offset between the base of the ski and theflexural neutral axis of the ski can be initially assumed to be zero,since the laminate thickness has not yet been defined.

The PhiZOffset function can be invoked for the purpose of offsetting aknown curvature profile to another elevation within the thickness of theski. For example, if a curvature profile is known along the base of theski (PhiB), and the elevation of the flexural neutral axis relative tothe base of the ski is known at every tangential position along thelength of the ski (Z_NA), then the PhiZOffset function could be used todetermine the corresponding curvature profile at the elevation of theflexural neutral axis of the ski (PhiNA).

At each tangential position along the length of the ski, the PhiZOffsetfunction calculates the radius of curvature at the initial elevation(R_(Phi1)) as the reciprocal of the curvature at the initial elevation(Phi1). The radius of curvature at the new offset elevation (R_(Phi2))is then calculated as: R_(Phi2)=R_(Phi1)−Z₂₁, where Z₂₁ is thedifference between the new elevation and the initial elevation. Thecurvature at the new offset elevation (Phi2) is then taken as thereciprocal of R_(Phi2). In cases where the absolute value of Phi1 isvery small (close to zero), the value of Phi2 is simply taken as beingequal to Phi1. The aforementioned procedure is carried out at everytangential position along the length of the ski in order to find theentire curvature profile at the new offset elevation.

The XmidZOffset function can be invoked for the purpose of offsetting atangential coordinate system (Xmid) to another elevation within thethickness of the ski. For example, if a tangential coordinate system isdefined along the base of the ski (XmidB), the curvature profile isknown along the base of the ski (PhiB), and the elevation of theflexural neutral axis relative to the base of the ski is known at everytangential position along the length of the ski (Z_NA), then theXmidZOffset function could be used to determine the correspondingtangential coordinate system that exists at the elevation of theflexural neutral axis of the ski (XmidNA).

At each node along the length of the ski, the XmidZOffset function findsa second node that is adjacent node of interest, but is closer than thenode of interest to the mid-base of the ski; the XmidZOffset functionthen calculates how the tangential distance between these two adjacentnodes will change at the new offset elevation. First, the tangentialdistance between these nodes (dXmid1) is calculated at the initialelevation. The average of the curvatures at the two nodes is thencalculated at the initial elevation (PhiAvg1). The average of the radiiof curvature at the two nodes (R_(PhiAvg1)) at the initial elevation isthen calculated as the reciprocal of PhiAvg1. The average of theelevation offset values at the two nodes is also calculated (Z_(21Avg)).The distance between the two nodes at the new offset elevation (dXmid2)is then calculated as:

${{dXmid}\; 2} = {\frac{{dXmid}\; 1\left( {R_{{PhiAvg}\; 1} - Z_{21\; {Avg}}} \right)}{R_{{PhiAvg}\; 1}}.}$

In cases where the absolute value of PhiAvg1 is very small (close tozero), the value of dXmid2 is simply taken as being equal to dXmid1.Starting at the mid-base of the ski and propagating outwards in bothdirections, the tangential coordinate system at the offset elevation isthen calculated by summing the relevant values of dXmid2 over the lengthof the ski.

Some calculations that are carried out by the ski design system 2000 mayrequire a geometric representation of the base surface of the ski.Conversely, some calculations that are carried out by the ski designsystem 2000 may require a geometric representation of the curvilinearsurface that passes through the flexural neutral axis of the ski at eachtangential position along its length. As such, the ski design system2000 can utilize the PhiZOffset function and the XmidZOffset function inorder to facilitate conversions between geometric representations of theski at these two different elevations within its thickness.

Operations that are carried out at the elevation of the base surface ofthe ski generally include: formulation of the initially assumed totaldeformed shape of the ski at the design point (ignoring torsionaldeformation); formulation of the initially assumed camber elevationprofile; and calculations pertaining to the snow penetration depthprofile, snow pressures, and snow loadings. Conversely, operations thatare carried out at the elevation of the flexural neutral axis of the skigenerally include: any calculations that involve addition and/orsubtraction of various curvature values (such as camber curvatures,flexural curvatures, rocker curvatures, and/or total curvatures);calculation of bending moment profiles and/or torsion moment profilesbased upon applied loadings; calculations that involve the relationshipbetween local flexural curvatures, local bending moments, and localflexural stiffnesses; and calculations that involve the relationshipbetween local rates of twist, local torsion moments, and local torsionalstiffnesses.

Strictly speaking, it may be more accurate to employ a third elevationcorresponding to a torsion axis (shear centre) of the ski when carryingout calculations pertaining to torsion. For most ski constructionarchitectures, the elevation of the torsion axis (shear centre) willlikely be very close to the elevation of the flexural neutral axis; assuch, the present example embodiment discussed herein assumes that thetorsion axis of the ski is located at the elevation of the flexuralneutral axis.

At 2010J, the ski design system 2000 initializes the geometry of theboot-binding system.

For example, the ski design system 2000 can first calculate thethree-dimensional geometry of the Cam1 camber elevation profile of theski. For example, the ski design system 2000 can invoke a functioncalled “CurveTwist2Shape” with PhiB set equal to the camber curvatureprofile and with thetaT set equal to thetaEloc_rel_init. TheCurveTwist2Shape function then returns the global (X, Y, and Z)coordinates and the v, b, and n orientation vectors of each node alongthe length of the base surface of the discretized ski when it exhibitsthe Cam1 camber elevation profile. Similar corresponding sets of global(X, Y, and Z) coordinates and v, b, and n orientation vectors arecalculated at the flexural neutral axis of the ski, as well as along theedge of the ski that is currently engaged with the snow.

The CurveTwist2Shape function can be invoked for the purpose ofdetermining the three-dimensional geometry of the ski as a function ofthe curvature profile along the base of the ski (PhiB) and the torsionangle profile (thetaT) over the length of the ski. This geometry iscalculated by assuming that each beam element adopts a helical shapethat is defined as a function of the local curvature and the local rateof twist exhibited by that element. The overall geometry of the ski isthen found by assuming a datum at the centre of the ski's effective baseregion (the mid-base point), and propagating outwards from this datum,one element at a time, following the helical geometry of each element.

Within each element, the helical geometry is defined as follows. Eachelement is bound by two nodes: one at its forward extremity and one atits rearward extremity. The bounding node that is closer to the mid-baseof the ski will be referred to as the inside node, whereas the boundingnode that is more distant from the mid-base of the ski will be referredto as the outside node. Each node corresponds to a tangential position(XmidB) along the length of the base of the ski, and the position ofeach node is expressed in terms of its global (X, Y, and Z) coordinates.The orientation of the base surface of the ski at each node isrepresented by three orthogonal direction vectors (orientation vectors)of unit length: the v vector is parallel to the local base contour ofthe ski and is aligned with the longitudinal tangential axis of the ski,and points rearward; the n vector is perpendicular to the base surfaceof the ski, and points in the direction that is opposite the snow(generally upwards); and the b vector is perpendicular to both the v andn vectors, and points toward the right hand side of the ski.

The total arc length of each helical segment (s_(h)) is defined as thedifference between the tangential coordinates (XmidB) of the nodes ateach end of the element of interest. The curvature of the element ofinterest (kappa_loc) is taken as the mean of the local curvatures (PhiB)found at the nodes at each end of the element. The rate of twist of theelement of interest (tau_loc) is taken as the difference between thelocal torsion angles (thetaT) found at the nodes at each end of theelement divided by the arc length (s_(h)) of the element. The helicalgeometry is defined as having an r_(h) coefficient that represents theradius of the cylindrical shell upon which the helical path exists. Thehelical geometry is also defined as having a c_(h) coefficient, where2πc_(h) represents the pitch of the helix (π≈3.1416), and where pitch isdefined as the distance between two adjacent loops of the helix measuredparallel to the helix axis (the central axis about which the helix pathcurves).

The coefficients c_(h) and r_(h) can be related to kappa_loc and tau_locas follows:

$\frac{c_{h}}{r_{h}} = {\frac{tau\_ loc}{kappa\_ loc}.}$

The rate of twist of the helical segment (tau_loc) can be related to thec_(h) and r_(h) coefficients as follows:

${tau\_ loc} = {\frac{c_{h}}{r_{h}^{2} + c_{h}^{2}}.}$

Combining the previous two expressions, the radius of the helical path(r_(h)) can be found as follows:

${r_{h} = \frac{1}{{kappa\_ loc}\left( {1 + \left( \frac{tau\_ loc}{kappa\_ loc} \right)^{2}} \right)}},$

where the value of r_(h) is always taken as positive. Similarly, thec_(h) coefficient of the helix can be found as follows:

${c_{h} = \frac{1}{{tau\_ loc}\left( {1 + \left( \frac{kappa\_ loc}{tau\_ loc} \right)^{2}} \right)}},$

where the value of c_(h) is taken as zero when tau_loc is zero.

The inclination angle of the helical path can then be round as:

${A_{h} = {\tan^{- 1}\left( {S_{kappa}\frac{c_{h}}{r_{h}}} \right)}},$

where S_(kappa) is taken as +1 if kappa_loc is positive and S_(kappa) istaken as −1 if kappa_loc is negative. In the case of zero torsion, thevalue of A_(h) would be zero and the helical path would simplify to acircular path. The angle of propagation through which the presentelement wraps around the helix axis can now be found as:

$\theta_{h} = {{- S_{kappa}}{\frac{s_{h}}{\left( {r_{h}^{2} + c_{h}^{2}} \right)^{1/2}}.}}$

A vector (j_(h)) that represents the orientation of the helix axis isfound by rotating the b vector at the inside node of the element aboutthe n vector at the inside node of the element, where said rotationangle is equal to A_(h). A vector (i_(h)) is defined as one that isperpendicular to j_(h) and tangent to the surface of the cylindricalshell upon which the helical path exists, at the position of the insidenode; this i_(h) vector is found by rotating the v vector at the insidenode of the element about the n vector at the inside node of theelement, where said rotation angle is equal to A_(h). A vector (k_(h))that is perpendicular to the base surface of the ski (thus pointingtoward the helix axis) at the position of the inside node of the elementis simply taken as being equal to the n vector at the inside node of theelement. Each of the i_(h), j_(h), and k_(h) vectors are then convertedinto unit vectors.

The magnitude of the displacement from the inside node of the element tothe outside node of the element along the directions of each of thei_(h), j_(h), and k_(h) vectors can be calculated as follows. Themagnitude of the displacement along the i_(h) vector is calculated asd_(ih)=S_(sh)(|r_(h) sin(θ_(h))|), where S_(sh) is taken as +1 if s_(h)is positive and S_(sh) is taken as −1 if s_(h) is negative. Themagnitude of the displacement along the j_(h) vector is calculated asd_(jh)=C_(h) θ_(h). The magnitude of the displacement along the k_(h)vector is calculated as: d_(kh) kappa (|r_(h)−r_(h) cos(θ_(h))|). Thevector distance from the inside node of the element to the outside nodeof the element is then calculated as follows: dXYZ=d_(ih)i_(h)+d_(jh)+d_(kh) k_(h).

The global (X, Y, and Z) coordinates of the outside node of the elementare then found by adding dXYZ to the global (X, Y, and Z) coordinates ofthe inside node of the element. The v vector of the outside node of theelement is found by rotating the v vector of the inside node of theelement about j_(h) by an angle of θ_(h). The b vector of the outsidenode of the element is found by rotating the b vector of the inside nodeof the element about j_(h) by an angle of θ_(h). The n vector of theoutside node of the element is found by rotating the n vector of theinside node of the element about j_(h) by an angle of θ_(h).

The global (X, Y, and Z) coordinates of the outside node of the elementare then adjusted slightly in order to account for the fact thattorsional deformations act about the torsion axis (shear centre) of theski's cross section, rather than at the elevation of the base of theski. This adjustment is made by: finding an offset vector (denoted by“NA_arm”) that is defined as the negative of the elevation of thetorsion axis (measured relative to the base surface of the ski)multiplied by the local n vector at the position of the outside node;rotating said NA_arm vector about the local v vector at the outside nodeby the negative of the local angle of twist of the element, wherein saidlocal angle of twist is calculated as the product of tau_loc and s_(h);subtracting the resulting rotated vector from the original NA_arm vectorin order to find a new vector (denoted by “dXYZ_NA_arm”) that representsthe negative of the displacement exhibited by the tip of the NA_armvector during said rotation; and adding this dXYZ_NA_arm vector to thepreviously calculated global (X, Y, and Z) coordinates of the outsidenode of interest. Recall that the present example embodiment discussedherein assumes that the torsion axis of the ski is located at theelevation of the flexural neutral axis of the ski.

Ultimately, for each node along the length of the ski, the global (X, Y,and Z) coordinates and the v, b, and n orientation vectors are found bybeginning at the mid-base of the ski and propagating outwards, elementby element, in accordance with the aforementioned procedure. TheCurveTwist2Shape function then returns the global (X, Y, and Z)coordinates and the v, b, and n orientation vectors of each node alongthe length of the base of the discretized ski. Once the CurveTwist2Shapefunction has returned the aforementioned data, the ski design system2000 can utilize the known geometry of the base of the ski to calculatean equivalent corresponding geometry at the elevation of the flexuralneutral axis of the ski, as well as an equivalent corresponding geometryat the position of the edge of the ski that is currently engaged withthe snow.

These geometries are determined by simply offsetting the global (X, Y,and Z) coordinates of the base nodes of the ski by known distances alongthe relevant local orientation vector (n vector for offsetting from theelevation of the base surface to the elevation of the flexural neutralaxis, or b vector for offsetting laterally to the position of the edge).At each of these offset nodes, the revised b orientation vector (b_(os))remains unchanged, and is simply set equal to b. The revised norientation vector (n_(os)) is then found at each of the offset nodes byfitting a vector between the global (X, Y, and Z) coordinates of theadjacent nodes (forward and aft of the node of interest) such that theresulting vector points rearward, converting this vector to a unitvector, and calculating the cross product of this unit vector and thelocal b_(os) orientation vector. The revised v orientation vector(v_(os)) is then calculated at each of the offset nodes as the crossproduct of the local b_(os) and n_(os) orientation vectors.

It is important to note that the aforementioned method that is employedby the CurveTwist2Shape function is not intended to be restrictive. Thepurpose of the CurveTwist2Shape function is to determine thethree-dimensional geometry of the ski as a function of the curvatureprofile along the base of the ski and the torsion angle profile over thelength of the ski; however, some embodiments may employ a multitude ofalternative analytical and/or computational methods in order to achievethat same purpose. Notwithstanding the foregoing, the CurveTwist2Shapefunction includes analytical and/or computational methods that areconducive for analyses of large deformations that may embody geometricnon-linearities.

The ski design system 2000 may then establish boot-binding loading pads.For example, the ski design system 2000 can invoke a function called“LoadingPads”.

The LoadingPads function assumes that the ski features two boot-bindingpad regions: one beneath the toe-cup of the binding, and one beneath theheel-cup of the binding. These pad regions do not necessarily have aphysical manifestation on the actual ski, but will serve as loadingregions where all of the binding loads will be analytically applied tothe ski model. Most on-piste carving skis and race skis feature abinding interface plate between the upper surface of the ski and thebinding system. Since all binding loads are applied to the ski throughsuch a binding interface plate, it is reasonable to define the geometryof the pad regions in a manner that observes the probable loadingenvironment between the binding interface plate and the ski.

For example, most modern binding interface plates for race skis areone-piece polymeric bars that are long enough to serve as mountingsurfaces for both the toe-cup and heel-cup of the binding system;however, these plates typically feature provisions to ensure that thereis adequate flexural (bending) compliance between the mounting points ofthe toe-cup and heel-cup (a flex element). As such, it is logical todefine the front boot-binding pad region as being positioned within thefootprint of the region of the binding interface plate that is ahead ofthe flex element, and to define the rear boot-binding pad region asbeing positioned within the footprint of the region of the bindinginterface plate that is aft of the flex element. Alternatively, someother types of binding interface systems are discretized into twoseparate plates: one plate is positioned beneath the toe-cup of thebinding, and another plate is positioned beneath the heel-cup of thebinding. In the case of such a two-piece binding interface system, itwould be logical to define the front boot-binding pad region as beingpositioned within the footprint of the front binding interface plate,and to define the rear boot-binding pad region as being positionedwithin the footprint of the rear binding interface plate.

Each pad region is characterized by a straight line (pad line) that isdrawn between the nodes that are nearest the forward-most and aft-mostextremities of the relevant pad region on the ski. The coordinates ofeach node of the ski that resides within the pad region is thenprojected onto this pad line in order to establish surrogate nodes thatreside on the pad line. Similarly, a new set of v_(pad), b_(pad), andn_(pad) orientation vectors is created at each node along the length ofthe pad region, wherein the v_(pad) vector is parallel to the pad line,the b_(pad) vector is equal to the b vector, and the n_(pad) vector iscalculated as the cross product of the v_(pad) and b_(pad) vectors.Since the position of the nodes on each pad line is dependent upon thedeformed or non-deformed geometry of the ski, it may be necessary toestablish the boot-binding pad regions of the ski under both camberedand loaded (design point) conditions, and it may be necessary to updatethese boot-binding pad regions at each iteration of the optimizationloop.

The ski design system 2000 can then initialize the positions of eachcomponent of the boot-binding system, measured with respect to a commondatum. For the convenience of the user, the INPUTS_Binding_Params inputfile can provide the relative positions of the various bindingcomponents with respect to various reference points within the bindinginterface plate 4200, the ski 3000, and the boot-binding system 4000. Inlight of these relative positions, it may be possible to recalculatesome of the initial binding component positions with respect to theelevation of the upper surface of the binding interface plate and thetangential coordinate system along the base of the ski (XmidB).

The ski design system 2000 can then initialize the geometry of thevarious components of the binding and plate system under initial camberconditions.

For example, the ski design system 2000 can begin by calculating theinitial assumed length of the toe-heel link of the binding system, as itis assumed that the Cam1 condition represents the circumstances underwhich the forward pressure system is set to its targeted initial preloadvalue. The length of the toe-heel link is calculated as the lineardistance between the mounting points of the toe-cup and heel-cup systemswith the ski geometry set to that of the Cam1 camber elevation profile.The ski design system 2000 can then proceed with invoking a functioncalled “BindingGeometry” with the ski geometry set to that of the Cam1camber elevation profile, and with the values of dZ_Cup_f and dZ_Cup_rinitially set to zero. It is initially assumed that the ski adopts itstargeted camber elevation profile regardless of whether or not a boot isinstalled in the binding; hence, Cam1 and Cam0 are initially identical.

The BindingGeometry function can be invoked for the purpose ofdetermining the geometric configuration that is adopted by the variouscomponents of the binding and plate system as a function of the skigeometry under specifically stipulated conditions (cambered ordeformed), as follows.

The INPUTS_Binding_Params input file provides the information necessaryto calculate the tangential coordinate along the base of the ski (XmidB)at which each of the following binding components and/or loading pointsis positioned: the mounting point of the toe-cup, the mounting point ofthe heel track, the position at which the toe-cup clamping force 4227(F_(Cup_f)) is applied, the position at which the toe-cup forwardpressure force 4225 (F_(FP_f)) is applied, the position of the toe-cuppivot axis 4210, the position at which the front (toe) AFD perch force4235 (F_(AFD_f)) is applied, the position at which the rear (heel) AFDperch force 4255 (F_(AFD_r)) is applied, and the position at which thebrake pedal force 4245 (F_(BP)) is applied. The INPUTS_Binding_Paramsinput file also provides the information necessary to calculate therelative elevation at which many of the binding components are mounted,measured with respect to the base of the ski, including: the position ofthe toe-cup pivot axis, the position of the heel-cup pivot axis, theposition at which the front (toe) AFD perch force 4235 (F_(AFD_f)) isapplied, and the position at which the rear (heel) AFD perch force 4255(F_(AFD_r)) is applied. The global (X, Y, and Z) coordinates of each ofthe aforementioned points is then found by finding the global (X, Y, andZ) coordinates that correspond to the tangential position of each ofthese points along the length of the base of the ski, and then addingthe product of the local n vector and the elevation of these pointsmeasured with respect to the base of the ski. The INPUTS_Binding_Paramsinput file also provides the length of the boot sole, the thickness ofthe toe lug of the boot, and the thickness of the heel lug of the boot.

It is initially assumed that the ski boot is rigid. As such, a vector(vSole) is drawn between the front and rear AFD perches; this vectorrepresents the flat surface of the sole 4140 of the boot 4100. Anothervector that represents the normal to the sole of the boot (nSole) iscreated by taking the cross product of vSole and the mean of the bvectors at each of the front and rear AFD perches. Using the known bootsole orientation and the known boot lug thicknesses, the BindingGeometryfunction then calculates the remaining unknown global (X, Y, and Z)coordinates of each of the load application points of the binding(F_(Cup_f), F_(Cup_r), F_(FP_f), F_(FP_r), F_(BP)); however, thiscalculation is carried out under the assumption that the boot sole isrigid.

Since boot soles are typically made from polymers that are quitestructurally compliant (soft), it is necessary to account for thedeformation of the boot lugs under the toe-cup and heel-cup clampingforces. As such, lug deflection vectors are defined at the toe lug 4120(dZ_Cup_f) and heel lug 4160 (dZ_Cup_r) of the boot, wherein both ofthese deflections are assumed to be parallel to the nSole vector, andact downward toward the snow. These lug deflection vectors are added tothe corresponding application points of the F_(Cup_f), F_(Cup_r),F_(FP_f), and F_(FP_r) forces. The values of dZ_Cup_f and dZ_Cup_r arecalculated outside of the BindingGeometry function; hence, theBindingGeometry function simply uses these values and assumes them to beaccurate.

Once the positions of the F_(Cup_f), F_(Cup_r), F_(FP_f), and F_(FP_r)force application points have been found, the BindingGeometry functionthen calculates the global (X, Y, and Z) coordinates of the heel-cuppivot axis 4270 as a function of the known global (X, Y, and Z)coordinates of the heel lug 4160, the known elevation of the heel-cuppivot axis 4270 above the base of the ski 3000, and the known geometryof the heel-cup 4260 (as provided in the INPUTS_Binding_Params inputfile). With the position of the heel-cup pivot axis 4270 now known, theBindingGeometry function is able to calculate the position of the heelcarriage 4280 within its heel track 4290.

The BindingGeometry function ultimately stores and returns all of theaforementioned geometric data pertaining to the boot-bindings system.The distance between the nominal mounting point of the heel track andthe true mounting point of the forward pressure spring (dXYZ_Mount_r) isthen calculated as a function of the present geometry (deformed ornon-deformed) of the ski and the length of the toe-heel link. If thepresent binding system does not include a toe-heel link, but it is usedin conjunction with a one-piece binding interface plate (wherein boththe toe-cup and heel-cup of the binding system are mounted to a commonplate), then said binding interface plate may itself be modeled tofunction as a toe-heel link for the binding system. Conversely, in theevent that the present binding system does not include a toe-heel linkand is not used in conjunction with a one-piece binding interface plate,then the value of dXYZ_Mount_r is set to zero. The effective position ofthe mounting point of the forward pressure spring (XYZ_CupMount_r) isthen calculated as the sum of dXYZ_Mount_r and the global (X, Y, and Z)coordinates of the nominal mounting point of the heel track 4290.

At 2020, the ski design system 2000 then begins running an optimizationloop, and iteratively repeats this loop until convergence is achieved.As depicted in FIG. 13, method 2020 can be summarized by the followingiteratively repeated sequence of operations: 2020A calculate the threedimensional deformed shape of the ski at the design point, assuming themost recently calculated profiles of total curvature and torsionaldeformation; 2020B calculate the snow loading distribution; 2020Ccalculate the binding geometry and binding loads at the design point;2020D calculate the bending moment profile, torsion moment profile,flexural curvature profile, and torsion angle profile at the designpoint; 2020E update the camber elevation profile of the unloaded ski,accounting for the influence of the boot-binding system; 2020F calculatethe required flexural stiffness profile, as necessary to achieve thedesired total curvature profile under the calculated loadingenvironment; 2020G calculate the required core thickness profile, andmake adjustments, as necessary, to address practical considerations;2020H calculate the actual flexural stiffness and torsional stiffnessprofiles of the ski, in light of the core thickness profile anduser-stipulated laminate architecture; 2020I adjust the camber curvatureprofile, as necessary, in order to compensate for regions that exhibitsub-optimal flexural stiffness, wherein said adjustments will typicallyconstitute the addition of rocker curvatures to regions that exhibitexcessive flexural stiffness; 2020J calculate the influence that theboot-binding system has upon the camber elevation profile of the ski;2020K calculate the torsional deformation of the ski at the designpoint, accounting for the latest ski design and most recently calculatedloadings; 2020L adjust the total curvature profile at the design point,in order to compensate for the effects of torsional deformation; 2020Madjust the width profile, in order to compensate for the effects oftorsional deformation; 2020N offset various curvature profiles andtangential coordinate systems between the elevations of the base of theski and the flexural neutral axis of the ski, as necessary, in order toensure that all curvature profiles and tangential coordinate systemscorrespond to the most recently calculated design specifications; and2020O check for convergence.

At 2020A, the ski design system 2000 begins by finding the deformedshape of the ski at the design point. For example, the deformed shape ofthe ski at the design point can be calculated by invoking theCurveTwist2Shape function with PhiB set equal to PhiTotalB and withthetaT set equal to thetaEloc_rel. The CurveTwist2Shape function thenreturns the global (X, Y, and Z) coordinates and the v, b, and norientation vectors of each node along the length of the base surface ofthe discretized ski under design point conditions. Similar correspondingsets of global (X, Y, and Z) coordinates and v, b, and n orientationvectors are calculated at the flexural neutral axis of the ski, as wellas along the edge of the ski that is currently engaged with the snow.

At 2020B, the ski design system 2000 calculates the penetration depth ofthe ski into the snow, as well as the resulting snow loadingdistribution along the length of the ski. For example, the penetrationdepth and snow loading distribution can be calculated using a functioncalled “SkiSnowInteraction”.

The SkiSnowInteraction function calculates the penetration depth andsnow loading distribution as follows. A ski of the calculated totaldeformed shape at the design point is modeled as being depressed into aflat surface of snow at the edging angle that has been stipulated at thedesign point, and the resulting local snow penetration depth iscalculated at each node along the length of the base of the ski; theresulting array (dSnow_geom) of local snow penetration depths representsthe overall snow penetration depth profile of the ski. The overallglobal penetration depth (distance between the mid-base point of the skiand the surface of the snow, measured perpendicular to the surface ofthe snow) and the pitch angle of the ski (angle between the surface ofthe snow and the v orientation vector that is positioned at the mid-basepoint of the ski, measured about an axis that lies parallel to thesurface of the snow and perpendicular to the longitudinal axis of theski) within the snow is adjusted using a pair of nested bracketedbisection search algorithms until the snow load distribution yields atotal snow force magnitude (F_(w_int)) that is equal to the normal forcethat the athlete imposes upon the ski under design point conditions(F_(nDP)), and the tangential position of the centroid of the snowloading (XmidBsnow) is equal to the user-stipulated position of thecentre-of-mass of the athlete. The aforementioned bracketed bisectionsearch algorithms are manifested as a pair of nested loops: the outerloop is responsible for finding the pitch angle of the ski, whereas theinner loop is responsible for finding the overall global penetrationdepth of the ski.

At each iteration of the inner loop, a function called “SnowLoading” isinvoked in order to find the total snow force magnitude (F_(W_int)) andthe tangential position of the centroid of the snow loading (XmidBsnow),and then the assumed global penetration depth of the ski is adjusted, asnecessary, in order to find the global penetration depth that will yielda total snow force magnitude that is equal to F_(nDP). At each iterationof the outer loop, a pitch angle is assumed, an initial globalpenetration depth is assumed, the inner loop is executed and run untilconvergence is achieved (F_(w_int) is approximately equal to F_(nDP),within an acceptable margin of error), and then the assumed pitch angleis adjusted, as necessary, in order to find the pitch angle that willcause the tangential position of the centroid of the snow loading(XmidBsnow) to be approximately equal to the user-stipulated position ofthe centre-of-mass of the athlete, within an acceptable margin of error.

Once overall convergence is achieved for both the pitch angle and theglobal penetration depth, the outputs from the most recently executedinstance of the SnowLoading function (w, f_(w), F_(w_int), andXmidBsnow) are stored for use in subsequent calculations. While theaforementioned SkiSnowInteraction function is presently described interms of the pitch angle of the ski and the global penetration depth ofthe ski, it is worth noting that in another embodiment, a similarfunction may be used wherein the ski remains stationary and the surfaceof the snow is translated and rotated with respect to the ski. Thisalternative function would, in fact, be less computationally expensivethan the function described here (since the many nodes and orientationvectors of the ski do not have to be translated and rotated in space).Nevertheless, the presently described function (wherein the ski itselfis translated and rotated) may be considered to be linked to the trueinteraction between a ski and snow in a more intuitive manner, and maytherefore be a more communicative means of describing how the functionworks.

The SnowLoading function can be invoked for the purpose of calculatingthe snow loading distribution that acts upon the base of the ski. A skiof a calculated total deformed shape at the design point is modeled asbeing depressed into a flat surface of snow at the edging angle that hasbeen defined at the design point, and the resulting local snowpenetration depth is then calculated at each node along the length ofthe base of the ski; the resulting array (dSnow_geom) of local snowpenetration depths represents the overall snow penetration depth profileof the ski.

A snow pressure distribution is calculated as a function of the snowpenetration depth profile (dSnow_geom) and the user-stipulatedconstitutive snow model. Some embodiments employ the constitutive snowmodel that is presented in Federolf_et_al_2010. This snow model is usedto calculate the average of the width-wise snow pressure distribution(p_(avg)) at the longitudinal position of each discretization point(node) along the length of the ski. Three coefficients are used torepresent the mechanical properties of the snow: A_(snow), B_(snow), andC_(snow), where the value of A_(snow) is dependent upon the local edgingangle of the ski.

The SnowLoading function begins at the tip (forward most point) of theski and migrates rearward, assessing the value of p_(avg) at each nodealong the length of the ski. A new variable (dSnow_max) is defined tostore the maximum of the local snow penetration depth values that havebeen observed in all of the nodes that are positioned forward of thenode at which p_(avg) is being assessed; the value of dSnow_max isinitially set to zero, and is adjusted as each node is assessed.Similarly, another new variable (p_(avg_max)) is defined to store themaximum of the p_(avg) values that have been calculated in all of thenodes that are positioned forward of the node at which p_(avg) is beingassessed; the value of p_(avg_max) is initially set to zero, and isadjusted as each node is assessed.

At the longitudinal position of each node, the local value of dSnow_geomis compared with the current value of dSnow_max. If the local value ofdSnow_geom exceeds the current value of dSnow_max, then the local valueof p_(avg) is calculated as: p_(avg)=A_(snow) d_(Snow)_geom+B_(snow),the value of dSnow_max is set equal to the local value of dSnow_geom,and the value of p_(avg_max) is set equal to the local value of p_(avg).Conversely, if dSnow_max exceeds the local value of dSnow_geom, then thelocal value of p_(avg) is calculated as: p_(avg)=p_(avg_max)+C_(snow)(dSnow_geom−dSnow_max), and the values of dSnow_max and p_(avg_max) areleft unchanged. In any case, if the local value of dSnow_geom is lessthan zero, then the local value of p_(avg) is set to zero. Similarly, ifthe local value of p_(avg) has been calculated to be less than zero,then the local value of p_(avg) is reset to zero.

At each nodal position along the length of the ski, a snow loading (w)is calculated as a force per unit length acting at the longitudinalposition of the corresponding node of the ski, and at the width-wiseposition of the centroid of the local width-wise snow pressuredistribution. The magnitude of this snow loading at each nodal positionis calculated as: w=p_(avg)W_(sub), where w_(sub) is the distancebetween the edge of the ski and the position on the base surface of theski that intersects the surface of the snow, measured width-wise(parallel to the local b orientation vector) along the base surface ofthe ski at the longitudinal position of the node of interest. The valueof W_(sub) can be calculated as follows:W_(sub)=(dSnow_geom)/sin(thetaEloc_eff), where thetaEloc_eff iscalculated as the absolute value of the inverse cosine of the dotproduct of the local n orientation vector and the surface normal of thesurface of the snow.

The width-wise snow pressure distribution acting upon the base of theski is determined at the longitudinal position of each node as afunction of the assumed constitutive snow model, the local edging angle,and the local snow penetration depth (dSnow_geom). The centroid of thiswidth-wise snow pressure distribution represents the width-wiseapplication point of the snow loading (w) at the longitudinal positionof the relevant node. This load application point is stored for eachcorresponding node along the length of the ski.

The SnowLoading function assumes that the width-wise snow pressuredistribution adopts a triangular geometry, wherein the snow pressure atthe width-wise position of the surface of the snow is zero, andincreases linearly to a value of 2 p_(avg) at the width-wise position ofthe edge of the ski (where the local snow penetration depth isachieved). In light of the aforementioned assumed triangular width-wisesnow pressure distribution, it can be reasoned that the centroid of thewidth-wise snow pressure distribution is positioned at a width-wiseposition on the base of the ski that is a distance of W_(sub)/3 from theedge, measured parallel to the local b orientation vector.

It is worth mentioning that this assumed triangular width-wise snowpressure distribution differs slightly from the width-wise snow pressuredistribution that is presented by Federolf_et_al_2010. While it wouldcertainly be possible to implement the width-wise snow pressuredistribution that is presented by Federolf_et_al_2010, the presentinventor has used his engineering judgment to reason that theaforementioned triangular width-wise snow pressure distribution embodiesa logical application of mechanical principals, and corroborates theempirical validation that was presented by Federolf_et_al_2010. It isalso worth mentioning at this point that alternative embodiments mayemploy completely different constitutive snow models (other than thatpresented by Federolf_et_al_2010), and the SnowLoading function may beadapted to work with such alternative constitutive snow models.

It is assumed that all snow pressure acts perpendicular to the contoursof the base of the ski (parallel to the local n orientation vectors).The resulting snow loading distribution (w) is stored as a force perunit length acting at the longitudinal position of each node of the ski,and the centroid of the width-wise snow pressure distribution is alsostored at the longitudinal position of each node of the ski. The snowloading distribution (w) is then converted into an array of equivalentelement forces (f_(w)), wherein each of these element forces is appliedto the mid-point of the relevant beam element of the ski, and has amagnitude that is equal to the length of the element multiplied by theaverage of the w values at each of the two nodes bounding the element.

Each row of the f_(w) array represents the elemental snow force at thelongitudinal position of the centroid of one element, and contains sixcolumns: the global (X, Y, and Z) Cartesian coordinates of the centroidof the local width-wise snow pressure distribution, which corresponds tothe application point of the relevant local snow force at thelongitudinal position of the mid-point of the element; and the threeorthogonal components of the local snow force vector, wherein said localsnow force vector has a direction that is perpendicular to the basesurface of the ski at the application point of the local snow force.

The vector sum of all snow forces (f_(w)) along the length of the ski iscalculated in order to find the total snow force (F_(w int)) that isacting upon the base of the ski. In addition, the tangential position ofthe centroid of the snow loading distribution (XmidBsnow) is calculatedas follows. A new orientation vector is defined as follows: v_(Fw)=[1,0, 0], where the aforementioned vector comprises X, Y, and Z orderedcomponents.

The SnowLoading function then calculates the sum of the moments of eachsnow force value in the f_(w) array about the origin (mid-base point),and the resulting moment vector is denoted by M_(Fw_int_0). Themagnitude of M_(Fw_int_0) is then calculated as the square root of thesum of the squares of its three vector components, and the resultingvalue is denoted by M_(Fw_0_Mag). A new unit vector (u_(MFw_0)) isdefined as having the same direction as M_(Fw_int_0), as follows: u_(M)_(Fw_0) =M_(Fw_int_0)/M_(Fw_0_Mag). A new direction vector is defined asfollows: V_(Fw_Arm)=F_(W_int)×M_(Fw_int_0), where x denotes a crossproduct operator. The magnitude of v_(Fw_Arm) is then calculated as thesquare root of the sum of the squares of its three vector components,and the resulting value is denoted by v_(Fw_Arm_Mag).

A new unit vector is then defined as follows:U_(Fw_Arm)=v_(Fw_Arm)/v_(Fw_Arm_Mag). The global (X, Y, and Z)coordinates of the centroid of the snow loading distribution are thencalculated as follows:

${{XYZ}_{w} = \frac{u_{{Fw}\_ {Arm}}M_{{{Fw}\_}0{\_ {Mag}}}}{u_{{{MFw}\_}0} \cdot \left( {u_{Fw\_ Arm} \times F_{w\_ int}} \right)}},$

where ⋅ denotes a dot product operator. A line (L_(Fw_int)) is definedhas having a direction that is parallel to the direction of theF_(w_int) force vector, while passing through the XYZ_(w) point. TheSnowLoading function then finds the global (X, Y, and Z) coordinates ofthe point at which the L_(Fw_int) line intersects the base surface ofthe ski; this intersection point is denoted by XYZ_(w_base). Finally,the tangential coordinate of the centroid of the snow loadingdistribution (XmidBsnow) is approximated as follows:XmidBsnow=v_(Fw)·XYZ_(w_base).

It would not be difficult to employ a slightly more accurate formulationfor XmidBsnow (perhaps by interpolating the XmidB position thatcorresponds to the global X component of the XYZ_(w_base) point);however, doing so would add computational expense to the SnowLoadingfunction, and the aforementioned approximation for XmidBsnow is quiteaccurate provided that the relative deflection of the ski is relativelysmall between the mid-base point and the XmidBsnow coordinate (which isalmost always the case). The SnowLoading function ultimately outputsvalues of w, f_(w), F_(w_int) and XmidBsnow.

At 2020C, the ski design system 2000 recalculates the orientation andposition of the boot-binding pad regions in the context of the skigeometry having been set to that of the deformed geometry that isexhibited at the design point. For example, the ski design system cancalculate the orientation and position of the boot-binding pad regionsusing the LoadingPads function. At 2020C, the ski design system 2000also calculates the geometric positions and loadings that are presentwithin the boot-binding system and the binding interface plate underdesign point loaded (deformed) conditions. For example, this can beachieved by employing a bracketed bisection search algorithm, asfollows.

Initial values of dZ_Cup_f and dZ_Cup_r are assumed. The BindingGeometryfunction is invoked with the ski geometry set to that present underdesign point loaded (deformed) conditions. A function called“BindingLoading” is then invoked using the output from theBindingGeometry function. The resulting toe and heel boot lug forces arethen calculated as a function of the user-stipulated toe and heel bootlug compliances (C_Lug_f and C_Lug_r) and the assumed toe and heel bootlug deflections (dZ_Cup_f and dZ_Cup_r), and these forces are comparedwith the corresponding toe-cup and heel-cup clamping forces that arefound (by the BindingLoading function) to be acting upon said boot lugs.The assumed values of dZ_Cup_f and dZ_Cup_r are adjusted as necessary,and the aforementioned search algorithm is repeated until convergencebetween boot lug forces and the corresponding toe-cup and heel-cupclamping forces is achieved. The pad loadings that result from thesebinding and plate loadings are then calculated by invoking a functioncalled “BindingPadLoading”.

The BindingLoading function can be invoked for the purpose ofdetermining the loadings that the binding and plate system imposes uponthe ski as a function of the geometric configuration that is exhibitedby the various components of the binding and plate system under specificconditions, as well as the snow loads that are being imposed upon thebase of the ski. If non-zero snow loads are present, then the sum of allelement snow forces (F_(w_int)) is calculated, and the moment of allelement snow forces (M_(w_int)) is taken about the position of the frontAFD perch. The position of each of the binding components is comparedwith its initial unloaded position in order to calculate the elasticresponse of the binding, and in turn, the loading that the boot-bindingsystem imposes upon the ski under this condition.

Binding loads include: toe-cup clamping force (F_(Cup_f)), heel-cupclamping force (F_(Cup_r)), forward pressure force at toe (F_(FP_f)),forward pressure force at heel (F_(FP_r)), normal force acting upon thefront AFD perch (F_(AFD_f)), normal force acting upon rear AFD perch(F_(AFD_r)), and brake pedal force (F_(BP)).

The forward pressure force at the heel (F_(FP_r)) is calculated as afunction of the difference between the position of the heel-cup pivotaxis under present conditions and that under Cam1 camber conditions,since this relative displacement constitutes a change in the length ofthe forward pressure spring relative to the conditions (Cam1) underwhich the forward pressure spring was adjusted to its targeted setting.The F_(FP_r) vector is defined such that it points in the same directionas the local v orientation vector at the position of the mounting pointof the heel-cup (v_(CupMount_r)).

The heel-cup clamping force (F_(Cup_r)) is calculated, in part, as afunction of the user-defined load-deflection response of the heel-cup4260, as well as the difference between the elevation of the heel-cup4260 under current conditions and relaxed conditions with no bootinstalled. In addition, since the F_(FP_r) force does not necessarilyact through the heel-cup pivot axis 4270, the presence of the F_(FP_r)force will generally cause a moment about the heel-cup pivot axis, whichwill manifest itself as an increase or decrease in the heel-cup clampingforce (F_(Cup_r)). As such, the total magnitude of F_(Cup_r) iscalculated as: F_(Cup_r)=n_(CupMount_r)(F_(Cup_r_s)+F_(FP_r_mag)C_(Cup_r_FP)), where F_(FP_r_mag) is the magnitude of the forwardpressure force at the heel, F_(Cup_r_s) is the magnitude of the portionof the heel-cup clamping force that is generated by the load-deflectionresponse of the heel-cup (excluding the effects of the moment thatF_(FP_r) generates about the heel-cup pivot axis), n_(CupMount_r) is thelocal n orientation vector at the position of the mounting point of theheel-cup, and C_(Cup_r_FP) is a coefficient that relates the forwardpressure force at the heel to the change in heel-cup clamping force thatis caused by the moment that F_(FP_r) generates about the heel-cup pivotaxis.

The value of C_(Cup_r_FP) is found as follows.

${C_{{Cup\_ r}{\_ FP}} = {- \frac{M_{{Axis\_ FP}{\_ r}} \cdot b_{CupMount\_ r}}{\left( {{XYZ}_{Axis\_ r} - {XYZ}_{Cup\_ r}} \right) \cdot v_{CupMount\_ r}}}},$

where b_(CupMount_r) is the local b orientation vector at the positionof the mounting point of the heel-cup, V_(CupMount_r) is the local vorientation vector at the position of the mounting point of theheel-cup, XYZ_(Axis_r) is the global (X, Y, and Z) Cartesian position ofthe heel-cup pivot axis, XYZ_(Cup_r) is the global (X, Y, and Z)Cartesian position of the heel-cup clamping force application point,M_(Axis_FP_r) represents the relationship between F_(FP_r) and themoment that F_(FP_r) generates about the heel-cup pivot axis, and ⋅denotes a dot product operator. The value of M_(Axis_FP_r) is calculatedas follows: M_(Axis_FP_r)=(XYZ_(FP_r)−XYZ_(Axis_r))×(V_(CupMount_r)),where XYZ_(FP_r) is the global (X, Y, and Z) Cartesian position of theapplication point of the forward pressure force at the heel, and xdenotes a cross product operator.

The toe-cup clamping force (F_(Cup_f)) is calculated, in part, as afunction of the user-defined load-deflection response of the toe-cup, aswell as the difference between the elevation of the toe-cup 4220 undercurrent conditions and relaxed conditions with no boot installed. Onceagain, since the forward pressure force does not necessarily act throughthe toe-cup pivot axis 4210, the presence of the forward pressure forcewill generally cause a moment about the toe-cup pivot axis 4210, whichwould manifest itself as an increase or decrease in the toe-cup clampingforce (F_(Cup_f)). As such, a new coefficient (C_(Cup_f_FP)) isintroduced as that which relates the forward pressure force at the toeto the change in toe-cup clamping force that is caused by the momentthat F_(FP_f) generates about the toe-cup pivot axis.

The value of C_(Cup_f_FP) is found as follows:

${C_{{Cup\_ f}{\_ FP}} = {- \frac{M_{{Axis\_ FP}{\_ f}} \cdot b_{CupMount\_ f}}{\left( {{XYZ}_{Axis\_ f} - {XYZ}_{Cup\_ f}} \right) \cdot v_{CupMount\_ f}}}},$

where b_(CupMount_f) is the local b orientation vector at the positionof the mounting point of the toe-cup, V_(CupMount_f) is the local vorientation vector at the position of the mounting point of the toe-cup,XYZ_(Axis_f) is the global (X, Y, and Z) Cartesian position of thetoe-cup pivot axis, XYZ_(Cup_f) is the global (X, Y, and Z) Cartesianposition of the toe-cup clamping force application point, andM_(Axis_FP_f) represents the relationship between F_(FP_f) and themoment that F_(FP_f) generates about the toe-cup pivot axis. The valueof M_(Axis_FP_f) is calculated as follows:M_(Axis_FP_f)=(XYZ_(FP_f)−XYZ_(Axis_f))×(−v_(CupMount_f)), whereXYZ_(FP_f) is the global (X, Y, and Z) Cartesian position of theapplication point of the forward pressure force at the toe.

The magnitude of the forward pressure force at the toe is thencalculated as:

${F_{{FP\_ f}{\_ mag}} = {- \frac{F_{{FP\_ f}{\_ UBvSole}} + {F_{{Cup\_ f}{\_ s}}\left( {{vSole} \cdot n_{CupMount\_ f}} \right)}}{\left( {{vSole} \cdot \left( {- v_{CupMount\_ f}} \right)} \right) + {C_{{Cup\_ f}{\_ FP}}\left( {{vSole} \cdot n_{CupMount\_ f}} \right)}}}},$

where n_(CupMount_f) is the local n orientation vector at the positionof the mounting point of the toe-cup, F_(Cup_f_s) is the magnitude ofthe portion of the toe-cup clamping force that is generated by theload-deflection response of the toe-cup (excluding the effects of themoment that F_(FP_f) generates about the toe-cup pivot axis), andF_(FP_f_UBvSole)=vSole·(F_(Cup_r)+F_(FP_r)+F_(w_int)).

Finally, the total toe-cup clamping force is calculated as:F_(Cup_f)=n_(CupMount_f) (F_(Cup_f_s)+F_(FP_f_mag) C_(Cup_f_FP)), andthe total forward pressure force at the toe is calculated as:F_(FP_f)=−v_(CupMount_f) F_(FP_f_mag).

In addition to the aforementioned binding loads, there also exists aload within the toe-heel link, which acts upon the toe-cup and heel-cupmounting structures at the elevation of the toe-heel link. The forcethat the toe-heel link imposes upon the heel-cup mount (F_(THL_r)) iscalculated as that which is necessary to prevent migration of the heelcarriage along the heel track, accounting for the presence of F_(Cup_r)and F_(FP_r), and recognizing that the axis of the toe-heel link is notnecessarily parallel to the heel track. The force that the toe-heel linkimposes upon the toe-cup mount (F_(THL_f)) is equal and opposite toF_(THL_r).

The brake pedal force (F_(BP)) is calculated as a function of theuser-defined load-deflection response of the brake pedal system and theposition of the brake pedal relative to relaxed (unloaded) conditions.It is assumed that F_(BP) acts parallel to nSole.

A moment imbalance about the front AFD perch (M_(UB_AFDf)) is thencalculated as the dot product of the mean of the local b vectors at thefront and rear AFD perches and the sum of the moments of F_(Cup_f),F_(Cup_r), F_(FP_f), F_(FP_r), F_(THL_f), F_(THL_r), F_(BP), andF_(w_int) about the front AFD perch.

The normal force acting upon the rear AFD perch is then calculated as:

${F_{AFD\_ r} = \frac{{nSole}*M_{U{B\_ AFDf}}}{{vSole} \cdot {dXYZ}_{AFDrf}}},$

where dXYZ_(AFDrf) is the difference between the global (X, Y, and Z)coordinates of the rear and front AFD perches. It is worth noting that,as a scalar operation, the aforementioned calculation of F_(AFD_r) issimply equal to M_(UB_AFDf) divided by the distance between the rear andfront AFD perches.

Finally, the normal force acting upon the front AFD perch (F_(AFD_f)) iscalculated as that which is required to equilibrate all of thepreviously calculated binding and snow load forces. Upon completion ofthe aforementioned boot-binding force calculations, the toe and heelboot lug forces are also calculated as a function of the assumed bootlug deflections (dZ_Cup_f and dZ_Cup_r) and the user-stipulated boot lugcompliances (C_Lug_f and C_Lug_r). The present geometry of the ski isalso used to calculate the relative positions of the front and rearregions of the binding interface plate under the relevant conditions. Ifthe user has provided data pertaining to the flexural stiffness of thebinding interface plate, then these data are used to determine thebending moment that is imposed upon the binding interface plate 4200,and in turn, the loading that the binding interface plate 4200 imposesupon the toe and heel boot-binding pads of the ski 3000.

The BindingPadLoading function can be invoked for the purpose ofconverting the forces and moments produced by the binding and platesystem into simplified distributed loads that are applied to theboot-binding pad regions. All of the toe binding loads (toe-cupclamping, toe forward pressure, front AFD perch, and front bindinginterface plate) are summed and converted into a force vector and momentvector positioned at the centroid of the front boot-binding pad region,and all of the heel binding loads (heel-cup clamping, heel forwardpressure, rear AFD perch, brake pedal, and rear binding interface plate)are summed and converted into a force vector and a moment vectorpositioned at the centroid of the rear boot-binding pad region.

These force and moment vectors are then converted into equivalentlinearly varying distributions of normal and shear forces over thesurfaces of the front and rear boot-binding pad regions, wherein saiddistributed loadings are represented by a normal load per unit lengthand a shear load per unit length that is applied between each adjacentpair of nodes on the pad line of the relevant boot-binding pad region.

The gradient (linear rate of change) of the normal force component ofthis distributed load (w_(BL) _(grad) ) is equal to 12 times the totalconcentrated moment imposed by the boot-binding system about the local bvector of the ski, divided by the cube of the length of the relevant padregion. The average value of the normal force component of thisdistributed load (w_(BL) _(avg) ) is equal to the total normal force(component of binding forces that act perpendicular to the surface ofthe relevant boot-binding pad region) divided by the length of therelevant pad region.

The normal force component of the distributed load can then becalculated at any position along the length of each pad as:w_(BL)=w_(BL) _(avg) +w_(BLgrad)x_(pad) _(c) , where x_(pad) _(c) is thetangential distance between the position of interest and the centroid ofthe relevant boot-binding pad region. The shear force component of thisdistributed load is assumed to be constant, and is therefore equal tothe total shear force (component of binding forces that act parallel tothe surface of the relevant boot-binding pad region) divided by thelength of the relevant pad region.

The distributed pad loading is then converted into an array ofequivalent element pad forces (f_(wPad)), wherein each of theseequivalent element forces is applied to the mid-point between twoadjacent nodes of the pad region, and has a magnitude that is equal tothe product of the distance between said adjacent nodes and the averageof the distributed binding load values corresponding to said adjacentnodes. Each row of the f_(wPad) array contains six columns: the global(X, Y, and Z) Cartesian coordinates of the longitudinal position of themid-point between two adjacent nodes of the pad region (the applicationpoint of the relevant equivalent element pad force); and the threeorthogonal components of the element pad force vector (along each of theglobal Cartesian axes).

At 2020D, the ski design system 2000 calculates the bending moment andtorsion moment profiles. For example, the bending moment and torsionmoment profiles can be carried out by a function called“BendingTorsion”.

The BendingTorsion function performs all calculations at the elevationof the flexural neutral axis of the ski; hence, the XmidNA tangentialcoordinate system is employed, and the global (X, Y, and Z) coordinatesand v, b, and n orientation vectors that are employed are those thathave been calculated at the elevation of the flexural neutral axis.

At each node along the length of the ski, the BendingTorsion functioncalculates the internal moment caused by the combination of the elementsnow forces (f_(w)) and the equivalent element pad forces (f_(wPad)).Each of these internal moments is calculated by finding all of theelement snow forces and equivalent element pad forces that act uponpoints that are positioned forward of the node of interest, and summingthe moments (cross products of moment arms and corresponding forces) ofall of these loads about the node of interest. The resulting internalmoment is represented by three orthogonal components, and the overallinternal moment profile due to f_(w) and f_(wPad) is stored as an arraythat is denoted by M₀.

In addition, the BendingTorsion function calculates the total momentimbalance (M_(UB)) by summing the moments (cross products of moment armsand corresponding forces) of each of the element snow forces (f_(w)) andeach of the equivalent element pad forces (f_(wPad)) about the origin(mid-base point) of the ski. In order to achieve a state of equilibrium,it may be necessary to add local concentrated moments at each of thefront and rear boot-binding pads. In addition, it is assumed that theboot exhibits infinite torsional stiffness; hence, since both of thefront and rear AFD perches are held firmly against the sole of the boot,it is assumed that both of these AFD perches share the same torsionangle (local edging angle). The addition of concentrated moments at eachboot-binding pad region serves two purposes: to equilibrate any globalmoment imbalances (M_(UB)), and to twist the ski (as necessary) suchthat the overall torsion angles exhibited at the longitudinal positionsof the front and rear AFD perches are equal to each other.

In the event that this is the first iteration of the optimization loopand the ski's flexural and torsional stiffness profiles have not yetbeen established, then it is assumed that the angles of twist are allequal to zero, and the necessary equilibrating concentrated momentsabout the longitudinal (torsion) axis of the ski are evenly distributedbetween the front and rear boot-binding pads.

The sum of the concentrated moment applied to the front boot-binding pad(M_(PF)) and the concentrated moment applied to the rear boot-bindingpad (M_(PR)) should be equal to M_(UB). It is therefore assumed that

${M_{PR} = {M_{osT} - \frac{M_{UB}}{2}}},$

and M_(PF)=−(M_(UB)+M_(PR)), where M_(osT) is the component of theconcentrated moment that is responsible for ensuring that the front andrear AFD perches share the same torsion angle. In order to ensure thatM_(osT) only contributes to the torsional deformation of the ski, anddoes not significantly alter the bending moment profile that is actingupon the ski, it may be important to ensure that M_(osT) acts about anaxis that passes through the front and rear AFD perches.

A bracketed bisection search algorithm is employed to find the values ofM_(PF) and M_(PR), as follows. A new direction vector (v_(osT)) isdefined as one that points from the front AFD perch to the rear AFDperch. Initial upper and lower bound values of M_(osT) are assumed asequal and opposite moment vectors, wherein both of these moment vectorsare parallel to v_(osT), and wherein the upper bound M_(osT) vectorpoints in the same direction as v_(osT). The current assumed value ofM_(osT) is the set equal to the mean of the currently assumed upper andlower bound values of M_(osT). The values of M_(PF) and M_(PR) are thencalculated as a function of M_(osT) and M_(UB), as described above.These moments are then applied to their respective boot-binding pads asuniformly distributed moments, wherein each moment is divided into nodalmoments that are applied to each of the nodes within the relevant padregion.

At each node, the internal moment that is caused by the concentrated padmoments (M_(PF) and M_(PR)) is calculated as the sum of all of the nodalpad moments that act at positions that are forward of the node ofinterest; the resulting internal moment profile due to the concentratedpad moments is denoted by M_(ConcPad). The total internal moment profile(M_(tot)) is then calculated as: M_(tot)=−(M₀+M_(ConcPad)), wherein saidtotal internal moment profile represents the total internal momentpresent at each node along the length of the ski.

The internal torsion moment at each node along the length of the ski iscalculated as the dot product of the total internal moment (M_(tot)) andthe local v vector. The rate of twist of each element is then calculatedas the local torsion moment divided by the local torsional stiffness.These rates of twist are then integrated over the length of the ski inorder to find the profile of relative torsion angles over the length ofthe ski. If the torsion angle at the front AFD perch is greater thanthat at the rear AFD perch, then the assumed lower bound value ofM_(osT) is set to the current value of M_(osT). Conversely, if thetorsion angle at the front AFD perch is less than that at the rear AFDperch, then the assumed upper bound value of M_(osT) is set to thecurrent value of M_(osT).

The aforementioned bracketed bisection search algorithm iterativelyrepeats the above-mentioned procedure until it converges upon a solutionwherein the torsion angles at the front and rear AFD perches areapproximately equal to each other within an acceptable margin of error.

The internal bending moment at each node along the length of the ski iscalculated as the negative of the dot product of the total internalmoment (M_(tot)) and the local b vector. The flexural curvature at eachnode is calculated as the local bending moment divided by the localflexural stiffness. The profile of local relative torsion angles is thenrotated about the v vector at the front AFD perch, wherein said rotationangle is equal to the negative of the calculated torsion angle at thefront AFD perch, thus setting the relative torsion angle to zero at thefront AFD perch. The resulting relative torsion angle profile(thetaTrel) is stored as an array; however, it should be noted thatthetaTrel does not account for the edging angle of the ski in itscurrent form.

At 2020E, the ski design system 2000 updates the initial camberelevation profile of the unloaded ski. For example, if the user hasindicated that the camber elevation profile is to be optimized withoutski boots installed, then Cam0 can be determined by invoking the CamProfile function; Cam1 can then be found by adding the flexuralcurvatures (bending curvatures) that are induced by the boot-bindingsystem 4000 under camber conditions to the Cam0 camber curvatureprofile. If the user has indicated that the camber elevation profile isto be optimized with ski boots installed, then Cam1 can be determined byinvoking the CamProfile function; Cam0 can then be found by subtractingthe flexural curvatures that are induced by the boot-binding system 4000under camber conditions from the Cam1 camber curvature profile.

In either case, it should be noted that curvatures may only be added orsubtracted at the elevation of the flexural neutral axis of the ski. Assuch, before flexural curvatures can be added or subtracted to cambercurvatures, it may first be necessary to offset the camber curvatureprofile to the elevation of the flexural neutral axis (by invoking thePhiZOffset function); once this addition or subtraction of curvatureshas been completed, the resulting curvature may then be offset back tothe base of the ski (by invoking the PhiZOffset function once again). Itshould also be noted that the aforementioned camber curvature profilemay initially only be created within the effective base length 3200 ofthe ski 3000. As such, the shovel nodes and tail nodes may be assigneduser-stipulated curvature values as necessary to create the upwardcurving geometries of these features.

At 2020F, the ski design system 2000 calculates the flexural curvatureprofile and the targeted flexural stiffness profile. For example,flexural curvatures can be calculated as the difference between thetotal curvature profile at the design point (at the elevation of theflexural neutral axis of the ski) and the Cam0 camber curvature profile(at the elevation of the flexural neutral axis of the ski). The targetedflexural stiffness profile can then be calculated as the bending momentat each node divided by the flexural curvature at each node (at theelevation of the flexural neutral axis of the ski).

At 2020G, the ski design system 2000 calculates of the core thicknessprofile. For example, by interpolating between the ski width versus corethickness versus stiffness response curves that were calculated prior toentering the optimization loop (see plot 5000 of FIG. 11 for a sample ofsuch response curves), the core thickness profile can be determined suchthat the targeted flexural stiffness profile is achieved. If any regionof the ski is assigned a core thickness that is less than theuser-stipulated minimum allowable core thickness, then these regions canbe assigned a core thickness that is equal to the user-stipulatedminimum allowable core thickness. This minimum allowable core thicknessis important as it may not be practicable to build a ski core thatexhibits a thickness profile that tapers to zero at any point along itslength.

A filleting routine can be employed such that any transitions betweenregions of constant core thickness and regions of variable corethickness are made smoothly, using a radial arc fillet. In someembodiments, other adjustments to the core thickness profile may beapplied, as necessary, in order to accommodate additional practicalconsiderations. The adjustments made to the core thickness profileexecuted at this stage of the optimization loop may ensure that allsubsequent operations that are carried out by the optimization loop areable to account for said adjustments.

At 2020H, the ski design system 2000 re-assesses the actual flexural andtorsional stiffness profiles of the ski at the optimization temperaturecontrol point. For example, the core thickness profile can be used inconjunction with the width profile and the user-stipulated skiconstruction (laminate architecture and material composition) tocalculate the flexural and torsional stiffness profiles that the skiwould exhibit with the most recently updated core thickness profile;this calculation can be carried out at each tangential coordinate alongthe length of the ski by invoking the EI_JG function.

At 2020I, the ski design system 2000 calculates of an overall camber androcker (Cam Rock) elevation profile.

For example, the camber and rocker profiles can be calculated asfollows. The elevation profile exists on a plane that is parallel to thelongitudinal axis of the ski and perpendicular to the base surface ofthe ski. Since the ski may now exhibit regions of sub-optimal flexuralstiffness (due to the need to maintain core thicknesses that are greaterthan or equal to the user-stipulated minimum, as well as other possibleadjustments that may have been made to the core thickness profile), theski may no longer achieve the targeted total curvature profile underdesign point conditions. As such, in regions of excessive flexuralstiffness, some rocker curvatures (reverse-camber curvatures) may needto be added to the existing camber curvature profile in order alleviatethe need for the ski to bend as much in said regions. Similarly, inregions of deficient flexural stiffness, some camber curvatures may needto be added to the existing camber curvature profile in order toincrease the amount by which the ski must bend in said regions.

The expected flexural curvature within each element of the ski (at theelevation of the flexural neutral axis) can be calculated as the bendingmoment within that element divided by the flexural stiffness of thatelement. A camber curvature adjustment profile (PhiCamAdj) can then becalculated as the total curvature at the design point minus the sum ofthe Cam0 camber curvature profile and the expected flexural curvatureprofile, where all of said curvature profiles are evaluated at theelevation of the flexural neutral axis of the ski. The aforementionedPhiCamAdj profile will typically comprise predominantly positivecurvature (rocker curvature) values in order to compensate for deficientflexural curvatures that are caused in regions of excessive flexuralstiffness; however, it should be noted that it is also possible for thePhiCamAdj profile to include some negative curvature (camber curvature)values, which could arise in the event that the ski exhibits someregions of deficient flexural stiffness.

Finally, the overall camber and rocker elevation profile (CamRockelevation profile) is represented by a CamRock curvature profile, whichcan be calculated as the sum of the camber curvature profile and thePhiCamAdj profile. CamRock0NA is defined as the sum of the Cam0 cambercurvature profile and the PhiCamAdj profile at the elevation of theflexural neutral axis of the ski. CamRock0B is then defined byoffsetting the CamRock0NA curvature profile to the elevation of the baseof the ski by invoking the PhiZOffset function. CamRock1NA is defined asthe sum of the CamRock0NA curvature profile and the flexural curvatureprofile that is induced by the boot-binding pad loads (PhiB_Binding_1)under unloaded (cambered) conditions. CamRock1B is then defined byoffsetting the CamRock1NA curvature profile to the elevation of the baseof the ski by invoking the PhiZOffset function. If this is the firstiteration of the optimization loop, then PhiB_Binding_1 is initiallyassumed to be a zero array, which will be populated later as theoptimization converges upon a design solution.

In light of the foregoing, it is important to recognize the significanceof the edging angle θ_(e) _(DP) that is stipulated at the design pointby the user. If the design point is defined using too shallow of anedging angle, then the ski design system 2000 may not provide asufficient amount of rocker at the tip and tail of the ski to facilitatea smooth snow loading distribution when edging angle values exceed thatwhich was stipulated at the design point. As a result, sharp peaks inthe snow loading distribution may be generated at the tip and tail whenedging angle values exceed that which was stipulated at the designpoint. Conversely, if the design point is defined using too steep of anedging angle, then the ski design system 2000 may provide an excessiveamount of rocker at the tip and tail of the ski, which may result indiminished edge engagement at the tip and tail when edging angles areshallower than that which was stipulated at the design point.

At 2020J, the ski design system 2000 calculates the influence that theboot-binding system has upon the camber elevation profile. For example,the ski design system 2000 can re-calculate the three-dimensionalgeometry of the most recently calculated CamRock1B elevation profile ofthe ski by invoking the CurveTwist2Shape function with PhiB set equal tothe CamRock1B curvature profile and with thetaT set equal tothetaEloc_rel_init. The CurveTwist2Shape function then returns theglobal (X, Y, and Z) coordinates and the v, b, and n orientation vectorsof each node along the length of the base surface of the discretized skiwhen it exhibits the CamRock1B elevation profile. Similar correspondingsets of global (X, Y, and Z) coordinates and v, b, and n orientationvectors are calculated at the flexural neutral axis of the ski, as wellas along the edge of the ski that is currently engaged with the snow.

The ski design system 2000 can then invoke the LoadingPads function inorder to recalculate the orientation and position of the boot-bindingpad regions in the context of the ski geometry having been set to thatof the most recently calculated CamRock1B elevation profile.

The ski design system 2000 can then update the effect that bindingforces have upon the CamRock1B elevation profile. The ski design system2000 begins by updating the length of the toe-heel link of the bindingsystem, as it is assumed that the Cam1 condition represents thecircumstances under which the forward pressure system is set to itstargeted initial preload value. The length of the toe-heel link iscalculated as the linear distance between the mounting points of thetoe-cup and heel-cup systems with the ski geometry set to that of themost recently calculated CamRock1B elevation profile.

The ski design system 2000 can then proceed with carrying out abracketed bisection search algorithm, as follows. Initial values ofdZ_Cup_f and dZ_Cup_r are assumed. The BindingGeometry function isinvoked with the ski geometry set to that of the most recentlycalculated Cam Rock1B elevation profile. The BindingLoading function isthen invoked using the output from the BindingGeometry function. Theresulting toe and heel boot lug forces are then calculated as a functionof the user-stipulated toe and heel boot lug compliances (C_Lug_f andC_Lug_r) and the assumed toe and heel boot lug deflections (dZ_Cup_f anddZ_Cup_r), and these forces are compared with the corresponding toe-cupand heel-cup clamping forces that are found (by the BindingLoadingfunction) to be acting upon said boot lugs. The assumed values ofdZ_Cup_f and dZ_Cup_r are adjusted as necessary, and the aforementionedsearch algorithm is repeated until convergence between boot lug forcesand the corresponding toe-cup and heel-cup clamping forces is achieved.

The pad loadings that result from these binding and plate loadings canthen be calculated by invoking the BindingPadLoading function. Finally,the structural influence that the binding forces have upon the ski underunloaded (camber) conditions can be calculated by invoking theBendingTorsion function (with element snow forces set to zero), whichcalculates the flexural curvature profile that is induced by theboot-binding pad loads (PhiB_Binding_1), evaluated at the elevation ofthe flexural neutral axis of the ski.

At 2020K, the ski design system 2000 calculates the torsionaldeformation of the ski at the design point. For example, using all ofthe latest information pertaining to the ski design, as well as the snowloading, binding loading, and binding interface plate loading that havemost recently been calculated under design point conditions, the skidesign system 2000 can calculate the relative twist angle profile(thetaTrel) by invoking the BendingTorsion function. The local edgingangle (thetaEloc_rel) at each discretization point (node) along thelength of the ski can then be re-calculated as the sum of thetaTrel andthetaEloc_rel_init. It should be noted that the value of thetaTrel isassumed to be zero at the positions of the front and rear AFD perches;as such, the values of thetaEloc_rel at the positions of the front andrear AFD perches will both be equal in magnitude to θ_(e) _(DP) .

At 2020L, the ski design system 2000 adjusts the total curvature profileat the design point (PhiTotalB) in order to partially compensate for theeffects of the torsional deformation of the ski. For example, a newarray (PhiTotalB_T) can be introduced to represent the assumed totalcurvature profile at the design point, which will be used forcomparative purposes when determining what is needed to compensate forthe effects of the torsional deformation of the ski. In someembodiments, PhiTotalB_T is set equal to the most recently calculatedPhiTotalB array. In other embodiments, PhiTotalB_T may be set equal toPhiTotalB_NoTwist in order to facilitate faster convergence of theoptimization solution; however, setting PhiTotalB_T equal to the mostrecently calculated PhiTotalB array will typically yield a moreprecisely optimized design solution that better accounts for theinherent geometric non-linearities of the present design problem.

The currently assumed deformed shape of the ski at the design point canthen be calculated by invoking the CurveTwist2Shape function with PhiBset equal to PhiTotalB_T and with thetaT set equal to thetaEloc_rel.With reference to a previously calculated dSnow_geom array, values ofthe previously calculated local snow penetration depths can be queriedat the positions of the front and rear AFD perches, and said values canbe denoted by dSnow_Bf and dSnow_Br for the front and rear AFD perches,respectively. By assuming the same triangular width-wise snow pressuredistribution that is employed in the SnowLoading function, it can bedetermined that the centroid of the local width-wise snow pressuredistribution at the longitudinal tangential position of the front AFDperch is positioned a distance of (dSnow_Bf)/(3 sin(thetaEloc_eff)) fromthe edge, measured parallel to the local b orientation vector, wherethetaEloc_eff is calculated as the absolute value of the inverse cosineof the dot product of the local n orientation vector and the surfacenormal of the surface of the snow. Similarly, it can be determined thatthe centroid of the local width-wise snow pressure distribution at thelongitudinal tangential position of the rear AFD perch is positioned adistance of (dSnow_Br)/(3 sin(thetaEloc_eff)) from the edge, measuredparallel to the local b orientation vector. The global position of theresulting centroid of the width-wise snow pressure distribution at thelongitudinal tangential position of the front AFD perch is denoted byXYZ_load_Bf. Similarly, the global position of the resulting centroid ofthe width-wise snow pressure distribution at the longitudinal tangentialposition of the rear AFD perch is denoted by XYZ_load_Br.

An imaginary vector (v_load) is fit between XYZ_load_Bf and XYZ_load_Br(pointing in the generally rearward direction); this v_load vector isthen converted into a unit vector. Another imaginary vector(b_horizontal) is defined as one that is horizontal and is perpendicularto the longitudinal axis of the ski at the centre of its effective baseregion (mid-base point), as follows: b_horizontal=[0, 1, 0], where theaforementioned vector comprises X, Y, and Z ordered components. Finally,a third imaginary vector (n_load) is defined as the cross product ofv_load and b_horizontal.

An imaginary plane (P_load) is then defined as one that is coincidentwith XYZ_load_Bf while having the surface normal n_load. It is thenassumed that the snow trace (the effective line of contact between theski and the snow) is defined by the intersection of this imaginaryP_load plane with the base contour of the ski. This effective snow tracegeometry is denoted by XYZ_load. The nominal plane of symmetry of theski (P_sym) is defined as a plane that is perpendicular to the basesurface of the ski (assuming zero torsional deformations), parallel tothe longitudinal axis of the ski, and passes through the longitudinalcentre-line of the ski at the position of its front AFD perch. As such,the P_sym nominal plane of symmetry is assumed to have a surface normalvector that is found as follows: n_(sym)=[0, cos(−θ_(e) _(DP) ),sin(−θ_(e) _(DP) )], where θ_(e) _(DP) is the edging angle of the ski atthe design point. XYZ_load is then projected onto this P_sym nominalplane of symmetry of the ski, and the resulting projected curve isdenoted by XYZ_surf_proj_b.

At each node along the length of the ski, the local curvature of theXYZ_surf_proj_b curve is determined by identifying an odd numberedseries of sequential nodes (no fewer than 3 nodes, and usually no morethan 7 nodes) that is centred about the node of interest, fitting acircular arc through this series of nodes, finding the radius of thiscircular arc, and then calculating the curvature as the reciprocal ofthis radius; the resulting curvature profile of the XYZ_surf_proj_bcurve is then stored as PhiTotB. The ski design system 2000 thenintroduces a flexural curvature offset array (PhiTorsionB) that will beemployed to partially compensate for the presence of torsionaldeformations; PhiTorsionB is calculated as the difference betweenPhiTotB and PhiTotalB_NoTwist.

A series of smoothing operations can then be applied to PhiTorsionB;failure to include these smoothing operations may result in divergentoscillations of the results generated at each iteration of theoptimization loop, which would ultimately prevent the ski design system2000 from finding a solution to the optimization problem. This series ofsmoothing operations begins by removing parcels of the PhiTorsionB datathat correspond to regions of the ski that exhibit discontinuities inthe mathematical functions that describe its curvature profile and/or itcore thickness profile (for example: the boot-binding pad regions; theupward curves within the shovel and tail regions; and the vicinity ofthe ski where the core thickness profile transitions from a variablethickness to a constant thickness). A cubic interpolation function isthen employed to re-populate the PhiTorsionB data that has been removed,thus ensuring that PhiTorsionB data exists at each discretization point(node) along the length of the ski. Finally, a Gaussian-weighted movingaverage smoothing function is applied to the PhiTorsionB data; thewindow size for this smoothing function is defined as a small percentage(usually between 3 percent and 5 percent) of the total number ofdiscretization points (nodes) within the effective base length of theski.

The total curvature profile at the design point (PhiTotalB) can then bere-calculated as the difference between PhiTotalB_T and PhiTorsionB.Ultimately, the aforementioned re-calculation of PhiTotalB serves toensure that the targeted snow trace geometry (that which is embodied bythe PhiTotalB_NoTwist total curvature profile in the absence oftorsional deformations) is achieved despite of the presence of non-zerotorsional deformations.

At 2020M, the ski design system 2000 adjusts the width profile in orderto partially compensate for the effects of the torsional deformation ofthe ski. In order to retain the longitudinal distribution of snowloadings (w) that had previously been calculated, it may be necessary toensure that the snow penetration depth profile is unaffected by theintroduction and/or modification of torsional deformations.

For example, the deformed shape of the ski at the design point can bere-calculated by invoking the CurveTwist2Shape function with PhiB setequal to PhiTotalB and with thetaT set equal to thetaEloc_rel. Withreference to a previously calculated dSnow_geom array, values of thepreviously calculated local snow penetration depths can be queried atthe positions of the front and rear AFD perches, and said values can bedenoted by dSnow_Bf and dSnow_Br for the front and rear AFD perches,respectively. It can be determined that the intersection of the basecontour of the ski and the surface of the snow at the longitudinaltangential position of the front AFD perch is positioned a distance of(dSnow_Bf)/(sin(thetaEloc_eff)) from the edge, measured parallel to thelocal b orientation vector, where thetaEloc_eff is calculated as theabsolute value of the inverse cosine of the dot product of the local norientation vector and the surface normal of the surface of the snow.Similarly, it can be determined that the intersection of the basecontour of the ski and the surface of the snow at the longitudinaltangential position of the rear AFD perch is positioned a distance of(dSnow_Br)/(sin(thetaEloc_eff)) from the edge, measured parallel to thelocal b orientation vector. The global position of the intersection ofthe base contour of the ski and the surface of the snow at thelongitudinal tangential position of the front AFD perch is denoted byXYZ_surface_Bf. The global position of the intersection of the basecontour of the ski and the surface of the snow at the longitudinaltangential position of the rear AFD perch is denoted by XYZ_surface_Br.

An imaginary vector (v_surface) is fit between XYZ_surface_Bf andXYZ_surface_Br (pointing in the generally rearward direction); thisv_surface vector is then converted into a unit vector. Another imaginaryvector (b_horizontal) is defined as one that is horizontal and isperpendicular to the longitudinal axis of the ski at the centre of itseffective base region (mid-base point), as follows: b_horizontal=[0, 1,0], where the aforementioned vector includes X, Y, and Z orderedcomponents. Finally, a third imaginary vector (n_surface) is defined asthe cross product of v_surface and b_horizontal. An imaginary plane(P_surface) is defined as one that is coincident with XYZ_surface_Bfwhile having the surface normal n_surface. This P_surface planerepresents the assumed surface of the snow.

At each discretization point (node), the ski design system 2000calculates the amount by which the width of the ski should be increasedsuch that the normal distance between the aforementioned imaginaryP_surface plane and the edge of the ski (measured perpendicular to theP_surface plane) is equal to the corresponding local value from thepreviously calculated dSnow_geom array; a new array (B_TorsionOffset) iscreated in order to store the required increase in width at eachtangential coordinate (XmidB) along the length of the base of the ski(negative values of B_TorsionOffset correspond to required decreases inwidth). The width profile of the ski is then updated by addingB_TorsionOffset to the previously assumed width profile of the ski. TheShovelTailWidths function is then invoked in order to re-assign localwidth profiles to the shovel nodes and tail nodes such that the planformgeometries of these regions satisfy the geometric relationships thatwere stipulated by the user.

At 2020N, the ski design system 2000 updates of the total curvatureprofile at the elevation of the flexural neutral axis of the ski(PhiTotalNA). For example, the ski design system can employ thePhiZOffset function to offset the PhiTotalB curvature profile by therelevant through-thickness distance. In addition, the ski design system2000 also updates the tangential coordinate system at the elevation ofthe flexural neutral axis (XmidNA). For example, the ski design system2000 can employ the XmidZOffset function to offset the XmidB coordinatesystem by the relevant through-thickness distance. At 2020N, the skidesign system 2000 also updates the tangential coordinate system at theelevation of the base of the ski under camber conditions (XmidBcam). Forexample, the ski design system 2000 can employ the XmidZOffset functionto offset the XmidNA coordinate system by the relevant through-thicknessdistance; this offsetting operation is carried out under both Cam0conditions (XmidBcam0) and Cam1 conditions (XmidBcam1).

At 2020O, the ski design system 2000 checks for convergence. Forexample, convergence can be held to have been achieved when the currentiteration of the optimization loop structure does not changesignificantly from the previous iteration of the optimization loopstructure. The following ski design characteristics can be checked forconvergence at the end of each iteration of the loop: snow penetrationdepth profile (dSnow_geom), core thickness profile, profile ofelevations of the flexural neutral axis, CamRock0NA curvature profile,Cam Rock1NA curvature profile, total curvature profile under designpoint loaded conditions, and deflected binding geometry under designpoint loaded conditions. If the change in any of these parametersexceeds acceptable limitations, then the ski design system 2000 canreturn to the beginning of the optimization loop structure at 2020A foran additional iteration.

Once convergence has been achieved, system 2000 exits the optimizationloop at 2030. As depicted in FIG. 14, the ski design system 2000 thencarries out method 2030 to prepare final design specifications of theoptimized ski. The method 2030 can be summarized by the followinggeneral sequence of steps: 2030A calculate the expected elasticspring-back; 2030B calculate the expected thermo-mechanical spring-back;2030C generate the recommended camber mould elevation profile; 2030Dcalculate the planform geometry of each ply of the laminate; 2030Ecalculate the expected elevation contour geometry of the topsheet of theski; 2030F format the design specification data for the output files;and 2030G write the output data files to memory.

At 2030A, the ski design system 2000 calculates the expected elasticspring-back that the ski will exhibit upon removal from themanufacturing tooling (mould) as a result of elastic strain energy thatis stored within each constituent of the ski construction (plies of thelaminate and solid components of the ski construction, such as edges,sidewalls, etc.).

For example, ski design system 2000 can begin by invoking the EI_JGfunction to recalculate the flexural stiffness of the complete ski atthe optimization temperature control point (EI_TOpt_tot). Ski designsystem 2000 can then calculate the local flexural stiffness of eachconstituent of the ski construction (plies of the laminate and solidcomponents of the ski construction, such as edges, sidewalls, etc.),wherein each of these calculations is carried out at the optimizationtemperature control point; these resulting local flexural stiffnessesare stored in an array (EI_TOpt_loc) containing data for eachconstituent of the ski at each tangential position along the length ofthe ski. The sum of the local constituent flexural stiffnesses(EI_TOpt_loc) can then be calculated and denoted by EI_TOpt_loc_sum.

For each constituent of the ski construction, the ski design system 2000can then calculate the local curvature that is imposed upon saidconstituent under CamRock0B conditions, wherein said curvature isevaluated at the elevation of the local flexural neutral axis of therelevant constituent; the resulting local curvatures are stored in anarray (PhiCamRock_loc) containing data for each constituent of the skiat each tangential position along the length of the ski. Ski designsystem 2000 can then calculate the elastic spring-back moments withineach constituent of the ski (momB_CamRock_loc) as the product ofEI_Topt_loc and PhiCamRock_loc. The total spring-back moment at eachtangential position (momB_CamRock_loc_sum) can then be calculated as thesum of the elastic spring-back moments within each constituent(momB_CamRock_loc). The expected elastic spring-back curvature of theski (PhiSpringBackNA) can then be calculated as the negative of the sumof elastic spring-back moments (momB_CamRock_loc_sum) divided by thedifference between the flexural stiffness of the complete ski(EI_TOpt_tot) and the sum of the flexural stiffnesses of eachconstituent of the ski (EI_TOpt_loc_sum).

At 2030B, the ski design system 2000 calculates the expectedthermally-induced spring-back (thermo-mechanical spring-back) that theski will exhibit upon removal from the manufacturing tooling (mould) asa result of processing-induced residual thermal strain energy that isgenerated within each constituent of the ski construction (plies of thelaminate and solid components of the ski construction, such as edges,sidewalls, etc.).

For example, at each of the temperature control points (TBuild, TOpt,and TCont), ski design system 2000 can calculate the longitudinalstiffness and flexural stiffness of the ski (using a method similar tothat employed by the EI_JG function).

In addition, at each of the temperature control points, system 2000 canalso calculate the thermal strains and curvatures of the full laminatein accordance with Classical Laminate Plate Theory (CLPT) using thesecant CLTE values that were calculated for each material as well as thetotal change in temperature between the cure temperature and therelevant temperature control point. The thermally-induced flexuralcurvature profile (thermo-mechanical flexural curvature profile) that isexpected to be exhibited at the optimization temperature control pointis stored in array that is denoted by PhiTherm_TOpt.

System 2000 can then calculate the camber curvature profiles that areexpected to be exhibited by the ski (at the elevation of its flexuralneutral axis) at each of the temperature control points(PhiCamRockNA_TBuild at the TBuild temperature, PhiCamRockNA_TOpt at theTOpt temperature, and PhiCamRockNA_TCont at the TCont temperature) asthe sum of the targeted CamRock0NA curvature profile and the differencebetween the thermo-mechanical flexural curvature profile that wascalculated at the relevant temperature control point and thethermo-mechanical curvature profile that was calculated at theoptimization temperature control point. Using the camber curvatureprofiles that were calculated at each temperature control point, system2000 can then calculate the corresponding expected camber elevationprofile at each temperature control point in Cartesian coordinates.

At step 2030C, the ski design system 2000 generates the recommendedcamber tooling (mould) elevation profile (the camber mould elevationprofile) that would ensure that the targeted camber elevation profile isachieved in light of the expected elastic spring-back andthermo-mechanical spring-back. The overall geometry of the cambertooling (mould), when viewed in elevation, is referred to as the “cambermould elevation profile”, wherein said elevation profile exists on aplane that is parallel to the longitudinal axis of the ski andperpendicular to the base surface of the ski. The “camber mouldcurvature profile” defines the curvature of the camber mould elevationprofile at any position along the length of the ski, wherein saidcurvature is measured within a plane that is parallel to thelongitudinal axis of the ski and perpendicular to the base surface ofthe ski.

For example, the camber mould curvature profile can be evaluated at theelevation of the flexural neutral axis of the ski (PhiCamRockNA_Mould)by calculating the difference between the CamRock0NA curvature profileand the sum of PhiSpringBackNA and PhiTherm_TOpt. The ski design system2000 may also enable a user to further modify the camber mould curvatureprofile by providing an optional empirical curvature offsetting array(CamRockB_Offset), which is stipulated at the elevation of the base ofthe ski. The PhiZOffset function can then be invoked in order to offsetthe CamRockB_Offset array to the elevation of the flexural neutral axisof the ski, resulting in a new empirical curvature offsetting arraydenoted by CamRockNA_Offset.

The CamRockNA_Offset curvature offsetting array is then added to thepreviously calculated PhiCamRockNA_Mould array. Finally, the cambermould curvature profile (PhiCamRock_Mould) is evaluated along thesurface contour of the camber mould by invoking the PhiZOffset functionto offset the PhiCamRockNA_Mould to the elevation of the base of theski. In addition, a tangential coordinate system along the surface ofthe mould (XmidBcam_Mould) is created by employing the XmidZOffsetfunction to offset XmidNA to the elevation of the base surface of theski, while observing the PhiCamRockNA_Mould curvature profile. Using thecamber mould curvature profile (PhiCamRock_Mould) and the correspondingtangential coordinates along the surface of the mould (XmidBcam_Mould),system 2000 calculates the recommended camber mould elevation profile inCartesian coordinates (XYZ_(Mould)).

At 2030D, the ski design system 2000 calculates the planform geometry ofeach ply of the laminate, assuming that it has been flattened, such thatit is representative of the geometry that said ply would exhibit beforeit has been laminated in a ski press.

For example, at each tangential coordinate along the length of the ski,the width of each ply can be calculated as a function of the total widthof the ski, the thickness of the core, and the user-stipulatedconstruction and cross sectional geometry of the ski. For eachtangential coordinate along the length of the base of the ski, thereexists a corresponding array of tangential coordinates at the mid-planeof each ply within the laminate; these tangential coordinates can becalculated for each ply of the laminate, as follows.

At each position (node point) along the surface of the camber mouldelevation profile (XYZ_(Mould)), system 2000 calculates the orientationof the local normal vector to the surface of the mould, and stores thiscollection of normal vectors in an array (n_(Mould)), wherein eachvector within the n_(Mould) array is of unit length. In addition, system2000 calculates the angles of inclination of the n_(Mould) vectorsrelative to the global Z-axis, measured about the global Y-axis, andstores the resulting angles in an array (θ_(nMould)). It is assumed thateach ply comprises a series of ply nodes and ply elements along themid-plane of said ply, wherein these ply nodes and ply elementscorrespond to the nodes and elements of the ski, respectively.

Each ply element is bound by ply two nodes: one at its forward extremityand one at its rearward extremity. The bounding ply node that is closerto the mid-base of the ski will be referred to as the inside ply node,whereas the bounding ply node that is more distant from the mid-base ofthe ski will be referred to as the outside ply node. Each ply nodecorresponds to a tangential position (XmidBcam_Mould) along the lengthof the camber mould elevation profile, and the position of each ply nodeis expressed in terms of its global (XYZ_(Ply)) coordinates. The localelevation of the mid-plane of the ply of interest (z_(Ply)), measuredperpendicular to the base surface of the ski, is calculated at eachtangential position as the sum of the thicknesses of all of the pliesbeneath the ply of interest, plus half of the thickness of the ply ofinterest. For the ply of interest, the global Cartesian coordinates ateach ply node are calculated as follows: XYZ_(Ply)=XYZ_(Mould)+n_(Mould)Z_(Ply).

Although each ply element is generally curved, the chord length of eachply element (C_(Ply_E)) can be calculated by finding the differencebetween the XYZ_(Ply) coordinates that correspond to the outer ply nodeand the XYZ_(Ply) coordinates that correspond to the inter ply node, andthen calculating the scalar magnitude of the resultant vector. The arclength of each ply element can then be calculated, as follows:

${S_{Ply\_ E} = {\frac{C_{Ply\_ E}d\; \theta_{nMould}}{2\; {\sin \left( \frac{d\; \theta_{nMould}}{2} \right)}}}},$

where dθ_(nMould) is the difference between the θ_(nMould) value thatcorresponds to the outer ply node and the θ_(nMould) value thatcorresponds to the inner ply node. A local tangential coordinate systemalong the length of the mid-plane of the ply of interest (XmidPly) isthen created by assuming a datum at the centre of the ski's effectivebase region (mid-base point), and propagating outwards from this datum,one ply element at a time. As such, the local tangential coordinate(XmidPly) that corresponds to each ply node is calculated as the sum ofthe lengths of all of the ply elements between the ply node of interestand the mid-base point of the ski, where the length of each ply elementis taken as S_(Ply_E).

Step 2030E of system 2000 calculates the expected elevation contourgeometry of the topsheet of the completed ski 3000, assuming that theski has been de-cambered onto a flat surface with an edging angle ofzero. These data may be valuable when designing a binding interfaceplate 4200 such that it closely conforms to the topsheet geometry of theski 3000.

At 2030F, the ski design system 2000 formats design specification datafor output files. It will be appreciated that the ski design system 2000can format a wide variety of different data.

For example, the design specification data may include the planformgeometry and thickness profile of each ply of the laminate, provided interms of the longitudinal tangential coordinate system of thecorresponding ply (XmidPly). The data may also include Engineering_Data,which summarizes engineering data pertaining to the profiles of thefollowing properties along the longitudinal tangential coordinates(XmidB) along the base of the ski: width, camber curvature, flexuralstiffness, torsional stiffness, and elevation of flexural neutral axis.In some embodiments, four versions of Engineering_Data can be created,representing the geometry and mechanical characteristics of the ski ateach of the three temperature control points (TBuild, TOpt, and TCont),as well as the target response (which, in theory, should be identical tothe response at the TOpt temperature control point). These engineeringdata may be used to carry out analytical and/or computationalsimulations to predict the expected performance of the ski design undervarious conditions, as well as to verify the accuracy of construction ofthe as-built skis.

In some embodiments, the design specification data may also includeCountours_of_Binding_Interface_Plates, which provides the elevationcontours of the topsheet of the ski (when fully de-cambered onto a flatsurface with an edging angle of zero), as well as the targeted elevationof the upper surface of the as-installed binding interface plate. Insome embodiments, the data may includeExtrapolated_Countours_of_Binding_Interface_Plates, which is similar to“Countours_of_Binding_Interface_Plates”, but includes the data coveringthe entire length of the ski.

In some embodiments, the design specification data may also includeMould_Surface_for_Camber_Profile, which provides detailed elevationgeometry data (global Cartesian coordinates and curvature values atcorresponding tangential positions) necessary to create the recommendedcamber mould elevation profile.

In some embodiments, the design specification data may also includeTarget_Camber_Profile, which provides detailed elevation geometry data(global Cartesian coordinates and curvature values at correspondingtangential positions) that describe the targeted CamRock0B elevationprofile. These camber elevation profile data can be used to verifyaccuracy of construction of the as-built ski.

At 2030G, the ski design system 2000 writes the output files formattedat 2030F to memory. The ski design system 2000 may be stored in avariety of formats, such as, but not limited to, text files (e.g., .txt,.csv, etc.), graphical files (e.g., .jpg, .tiff, etc.), and/orcomputer-aided design (CAD) or computer-aided manufacturing (CAM) files.

It will be appreciated that the ski design system 2000 cansimultaneously account for flexural deformations and torsionaldeformations, while still achieving the precise snow trace geometry andsnow pressure distribution that are targeted during the initializationphase of the solution. This is made possible by the unique processingoperations and sequencing of the processing operations that are employedby the ski design system 2000. As described above, the ski design system2000 assumes zero torsional deformation for the purpose of initiallydefining the approximate deformed shape (only flexural deformations) andwidth profile that would be necessary to achieve a targeted snow tracegeometry and snow loading distribution; the ski design system 2000 thenproceeds with designing a ski that would achieve said flexuraldeformation; the ski design system 2000 then finds the total deformationthat the resulting ski design would exhibit, while accounting for bothbending and torsion; the ski design system 2000 then makes adjustmentsto compensate for the calculated torsional deformation, while ensuringthat the resulting snow loading distribution remains essentiallyunchanged; the ski design system 2000 then repeats the aforementionedprocesses until it has converged upon a stable design solution.Adjustments that are made to compensate for the effects of non-zerotorsional deformation are carried out in a manner that ensures that theresulting snow loading distribution remains essentially unchangedbetween successive iterations of the optimization loop; resulting insuccessful convergence of the optimization solution.

By initially neglecting the effects of torsional deformation and thenlater making adjustments to account for the presence of non-zerotorsional deformation, the optimization problem is dramaticallysimplified, and it becomes possible to reliably achieve convergence evenunder highly non-linear conditions (such as: use of non-linearconstitutive snow models and/or inclusion of geometric non-linearitiesdue to large deformations of the ski). This simplified solution to theoptimization problem also facilitates the inclusion of numerousfidelity-improving details (such as a non-linear constitutive snowmodel, complex boot-binding loadings, etc.), which might otherwise beprecluded from inclusion in the optimization solution due to logisticalchallenges.

With the ski design system 2000, it may be possible for a user tostipulate an exact snow trace geometry and snow pressure distribution.The ski design system 2000 can then rapidly find a design solution thatwill precisely satisfy these optimization criteria.

It should be noted that the ski design system 2000 is most applicable todesigns that are dominated by flexural deformation. A ski that exhibitstotal deformations that comprise substantial torsional compliance maypresent a challenging optimization problem for the ski design system2000; hence, extremely wide skis and/or snowboards may constitute morechallenging optimization problems. Nevertheless, the ski design system2000 may be able to generate designs for skis having waist widths thatare near the upper limit (90 mm) of what is generally consideredacceptable for on-piste carving skis. Also, for wider skis andsnowboards, the ski design system 2000 may be able to achieveconvergence at an acceptable rate provided that slightly more generousconvergence tolerances are employed.

It should be appreciated that the ski design system 2000 can employ anadjustment to the camber curvature profile in order to compensate forregions that exhibit sub-optimal flexural stiffness. This need to adjustthe camber curvature profile may arise, for example, as a result of thefact that it is not practicable to build a ski that exhibits a flexuralstiffness profile that tapers to zero at the tip and tail of the ski,thus leading to the existence of regions of the ski that exhibitexcessive flexural stiffness; in this case, it may be necessary to addstrategically placed rocker to some regions of the camber curvatureprofile.

It will also be appreciated that ski design system 2000 canstrategically define a camber elevation profile for the purpose ofachieving a nearly optimal snow trace geometry and snow loadingdistribution at two distinct design points; in effect, the ski designsystem 2000 can design the camber elevation profile of a ski such thatthe ski will achieve a desired relationship between ski-snow interactionand edging angle.

It is generally understood that most alpine race skis and on-pistecarving skis exhibit camber geometries that are defined for two primarypurposes: to ensure that snow loads are distributed over the length ofthe ski in a desired manner, and to prescribe the amount of strainenergy that is to be stored in the ski during each carved turn. Whilethe aforementioned characteristics are, in fact, directly linked to thecamber elevation profile of a ski, the present inventor does not believethat they, alone, should drive the design process for the camberelevation profile. The manner in which the ski-snow interaction variesas a function of edging angle is also directly linked to the design ofthe camber elevation profile, and the present inventor believes thatthis characteristic should be treated with importance when designing thecamber elevation profile of alpine race skis and on-piste carving skis.As such, the ski design system 2000 can define the camber elevationprofile primarily as a function of this ski-snow interaction versesedging angle relationship. It should be noted, however, that in someembodiments, the inputs of the ski design system 2000 can be adjusted inorder to arrive upon a compromise between a desired camber height (forexample, for the purpose of achieving a desired amount of strain energyin a given turn) and a desired ski-snow interaction verses edging anglerelationship.

In some embodiments, a mathematical optimization software applicationmay be configured to perform one or more of the steps of methods 100,200, 2010, 2020, and 2030.

In some embodiments, the finite element method can be used to performone or more of the steps of methods 100, 200, 2010, 2020, and 2030. Thatis, the finite element method may be used by the ski design system 2000to provide structural analysis and/or snow modeling. For example, thefinite element method may be used to calculate the three-dimensionalshape of the ski at 2020A, calculate the snow loading distribution at2020B, calculate the binding geometry and binding loads at 2020C,calculate the bending and torsion moments at 2020D, etc. The finiteelement method may consider the effects of three-dimensional strains,including, but not limited to: transverse shear, longitudinal bendingabout the n-axis, and local lateral bending about the v-axis. As such,use of the finite element method may facilitate more accurate analysesof ski construction architectures that include extremely compliant(soft) materials, such as viscoelastic and/or elastomeric damping layersthat may be included within the laminate. In some embodiments, it may benecessary to create a rigid representation of the ski in its assumeddeformed shape in order to facilitate an initial calculation of the snowloads that are imposed upon the ski.

In some embodiments, multibody simulation (such as multibody dynamicssoftware) can be used to perform one or more of the steps of methods100, 200, 2010, 2020, and 2030. That is, multibody simulation may beused by the ski design system 2000 to provide structural analyses of theski and/or boot-binding system. For example, the ski can be representedby a series of rigid bodies that are connected by springs havingcalibrated stiffnesses, such that the overall flexural and torsionalresponse of this chain of rigid bodies and springs approximates theintended flexural and torsional response of the ski.

In some embodiments, a hybrid of various analytical and/or computationalmethods can be used to perform one or more of the steps of methods 100,200, 2010, 2020, and 2030. In some of such cases, it may be beneficialto employ a combination of both the finite element method and multibodysimulation techniques in order to assess the behaviour of the ski andboot-binding systems. For example, in some embodiments that employ thefinite element method to model the behaviour of the ski, theboot-binding system can be modeled using multibody simulation, and theresulting forces and moments that are calculated by said multibodysimulation can then be applied to the finite element model of the ski.

In some embodiments, the boot-binding system can be represented by amore simplistic system of forces and moments. In these embodiments, itmay be unnecessary to collect information (geometry, load-deflectionresponse, etc.) pertaining to each individual component of theboot-binding system; instead, experiments can be carried out in order todirectly ascertain how the boot-binding system affects the flexuralresponse of the ski, and this effect is imposed upon the ski as afunction of the curvature of the ski and/or the relative angle betweenthe mounting points of the toe-cup and heel-cup of the binding system.

For example, a surrogate ski of known flexural stiffness can beinstrumented with a series of load transducers fitted between the uppersurface of the ski and the lower surface of the binding system; as thissurrogate ski is deformed, the loads measured by the load transducerscan be measured as a function of the relative angle between the mountingpoints of the toe-cup and heel-cup of the binding system, oralternatively, as a function of the flexural curvature that is exhibitedby the ski within the vicinity of the boot-binding system. Uponcompletion of this test, it would then be possible to describe the loadsthat the boot-binding system imposes upon the ski as a function of therelative angle between the mounting points of the toe-cup and heel-cupof the binding system, or alternatively, as a function of the flexuralcurvature that is exhibited by the ski within the vicinity of theboot-binding system.

As described above, the inputs of the ski design system 2000 can includesome quantitative values that help to describe the skiing technique thatis employed by the athlete, such as: the targeted position of thecentre-of-mass of the athlete along the length of the ski; values ofθ_(e) _(DP) , θ_(e) _(CP1) , and θ_(e) _(CP2) ; and data to ascertainthe values of F_(nDP), F_(nCP1), and F_(nCP2). In some embodiments, someor all of the aforementioned input data can be collected by way of aninstrumented experimental test, as follows. An existing pair of skisthat is similar to that which is to be designed can be equipped with aportable electronic data acquisition system (data logger) that isconnected to a few small instruments and/or sensors, such as: a triaxialaccelerometer, a triaxial gyroscope (such as a vibrating structuregyroscope, or Coriolis vibratory gyroscope), and an array of pressuresensors and/or force sensors (such as load cells and/or thin-film forcesensitive resistors) mounted in the vicinity of the boot-binding system.The aforementioned instrumentation may be integrated into a bindinginterface plate system, such that the athlete is not able to perceive(or be disturbed by) its presence while using said skis.

With the data logger collecting data from each of the aforementionedinstruments at some predetermined sampling rate, the athlete would thenski on said instrumented skis under conditions that are similar to thosefor which a new ski design is to be optimized by system 2000. The datathat is collected by each instrument may then be smoothed and/orfiltered in order to remove noise, and then the resulting data could beprocessed in order to ascertain meaningful metrics that correspond toeach sample time of the logged data; such metrics could include, butwould not be limited to: the position of the centre-of-mass of theathlete along the length of the ski, the edging angle of the ski(θ_(e)), and the component of the total force that acts perpendicular tothe base surface of the ski (F_(n)).

A two-dimensional plot could be created, wherein each F_(n) value isplotted against the absolute value of the tangent of the correspondingθ_(e) value. FIG. 15 depicts a fabricated hypothetical example of such aplot 6000. A simple linear regression trend line can be fitted throughthe resulting data plot; said trend line could be described by anequation of the form: F_(n)=b_(tl)+m_(tl)|tan(θ_(e))|, where m_(tl) isthe slope of the trend line, and b_(tl) is the intercept of the trendline with the F_(n)-axis. A user could select a value of θ_(e) _(DP) ,and the corresponding value of F_(nDP) could be calculated as:F_(nDP)=b_(tl)+m_(tl)|tan(θ_(e) _(DP) )|. Similarly, a user could selecta value of θ_(e) _(CP1) , and the corresponding value of F_(nCP1) couldbe calculated as: F_(nCP1)=b_(tl)+m_(tl)|tan(θ_(e) _(CP1) )|. Finally, auser could select a value of θ_(e) _(CP2) , and the corresponding valueof F_(nCP2) could be calculated as: F_(nCP2)=b_(tl)+m_(tl)|tan(θ_(e)_(CP2) )|.

Referring to FIG. 15, it is expected that F_(n) values that correspondto edging angles (θ_(e)) that are close to zero will likely exhibitinclement levels of scatter; this expectation is due, in part, to thefact that an athlete typically conducts a broad variety of maneuverswith his/her skis oriented approximately flat on the snow. When anathlete is standing still on his/her skis on level terrain, the measurededging angles (θ_(e)) will be close to zero, and the corresponding F_(n)values measured by each ski will be approximately equal to half of thestatic weight of the athlete.

Conversely, when linking multiple carved turns, an athlete often brieflyunloads his/her skis while transitioning between the right and leftedges of his/her skis; consequently, a ski that is in the process oflinking multiple carved turns may undergo relatively low F_(n) values(less than half of the static weight of the athlete) as its orientationpasses through edging angle that are close to zero.

In order to alleviate the influence of inclement scatter exhibited bythe F_(n) values that correspond to edging angles (θ_(e)) that are closeto zero, in some embodiments, the aforementioned simple linearregression trend line may be only fitted through data that correspond toedging angle magnitudes that are greater than some minimum value. Thismay help to ensure that said trend line is only representative of carvedturns, and is not adversely influenced by other maneuvers. For example,plot 6000 has been constructed with said linear regression trend linefitted through data that correspond to |tan(θ_(e) _(CP1) )| values thatare greater than unity.

It will be appreciated that numerous specific details are set forth inorder to provide a thorough understanding of the example embodimentsdescribed herein. However, it will be understood by those of ordinaryskill in the art that the embodiments described herein may be practicedwithout these specific details. In other instances, well-known methods,procedures and components have not been described in detail so as not toobscure the embodiments described herein. Furthermore, this descriptionand the drawings are not to be considered as limiting the scope of theembodiments described herein in any way, but rather as merely describingthe implementation of the various embodiments described herein.

It should be noted that terms of degree such as “substantially”, “about”and “approximately” when used herein mean a reasonable amount ofdeviation of the modified term such that the end result is notsignificantly changed. These terms of degree should be construed asincluding a deviation of the modified term if this deviation would notnegate the meaning of the term it modifies.

In addition, as used herein, the wording “and/or” is intended torepresent an inclusive-or. That is, “X and/or Y” is intended to mean Xor Y or both, for example. As a further example, “X, Y, and/or Z” isintended to mean X or Y or Z or any combination thereof.

It should be noted that the term “coupled” used herein indicates thattwo elements can be directly coupled to one another or coupled to oneanother through one or more intermediate elements. Furthermore, the term“body” typically refers to the body of a patient, a subject or anindividual who receives the ingestible device. The patient or subject isgenerally a human or other animal.

The embodiments of the systems and methods described herein may beimplemented in hardware or software, or a combination of both. Theseembodiments may be implemented in computer programs executing onprogrammable computers, each computer including at least one processor,a data storage system (including volatile memory or non-volatile memoryor other data storage elements or a combination thereof), and at leastone communication interface. For example and without limitation, theprogrammable computers (referred to herein as computing devices) may bea server, network appliance, embedded device, computer expansion module,a personal computer, laptop, personal data assistant, cellulartelephone, smartphone device, tablet computer, a wireless device or anyother computing device capable of being configured to carry out themethods described herein.

In some embodiments, the communication interface may be a networkcommunication interface. In embodiments in which elements are combined,the communication interface may be a software communication interface,such as those for inter-process communication (IPC). In still otherembodiments, there may be a combination of communication interfacesimplemented as hardware, software, and combination thereof.

Program code may be applied to input data to perform the functionsdescribed herein and to generate output information. The outputinformation is applied to one or more output devices, in known fashion.

Each program may be implemented in a high level procedural or objectoriented programming and/or scripting language, or both, to communicatewith a computer system. However, the programs may be implemented inassembly or machine language, if desired. In any case, the language maybe a compiled or interpreted language. Each such computer program may bestored on a storage media or a device (e.g. ROM, magnetic disk, opticaldisc) readable by a general or special purpose programmable computer,for configuring and operating the computer when the storage media ordevice is read by the computer to perform the procedures describedherein. Embodiments of the system may also be considered to beimplemented as a non-transitory computer-readable storage medium,configured with a computer program, where the storage medium soconfigured causes a computer to operate in a specific and predefinedmanner to perform the functions described herein.

Furthermore, the system, processes and methods of the describedembodiments are capable of being distributed in a computer programproduct comprising a computer readable medium that bears computer usableinstructions for one or more processors. The medium may be provided invarious forms, including one or more diskettes, compact disks, tapes,chips, wireline transmissions, satellite transmissions, internettransmission or downloadings, magnetic and electronic storage media,digital and analog signals, and the like. The computer useableinstructions may also be in various forms, including compiled andnon-compiled code.

Various embodiments have been described herein by way of example only.Various modification and variations may be made to these exampleembodiments without departing from the spirit and scope of theinvention, which is limited only by the appended claims.

I claim:
 1. A computer-implemented method for generating a design for agliding board, the method comprising operating a processor to: define adesired carved turn of the gliding board, the desired carved turn beingdefined at least by a nominal edging angle and an athlete load profile,wherein the athlete load profile represents a load that is applied by anathlete to the gliding board during the desired carved turn; define adesired global curvature profile, wherein the desired global curvatureprofile corresponds to a desired snow trace profile for the desiredcarved turn; generate a desired deformed shape of the gliding boardduring the desired carved turn, the desired deformed shape of thegliding board being defined at least by a desired total curvatureprofile, wherein the desired total curvature profile is initially set tocorrespond to the desired global curvature profile; generate a sidecutprofile of the gliding board; generate a width profile of the glidingboard based at least on the sidecut profile; generate a camber profileof the gliding board; generate at least one stiffness design variableprofile, wherein the at least one stiffness design variable profile, inconjunction with at least the width profile and at least one glidingboard material property, dictates a resulting flexural stiffness profileand a resulting torsional stiffness profile of the gliding board;generate a total load profile based at least on the athlete loadprofile, wherein the total load profile represents a total load that isapplied to the gliding board during the desired carved turn, and whereingenerating the total load profile comprises generating a desired snowpenetration depth profile; modify at least the width profile, thesidecut profile and at least one of the at least one stiffness designvariable profile at least once by: calculating a desired flexuralstiffness profile of the gliding board based at least on the total loadprofile, such that the desired flexural stiffness profile approximatelyachieves the desired total curvature profile during the carved turn;modifying at least one of the at least one stiffness design variableprofile such that the resulting flexural stiffness profile isapproximately equal to the desired flexural stiffness profile;calculating a torsional deformation profile of the gliding board duringthe carved turn based at least on the total load profile; modifying thedesired total curvature profile based at least on the torsionaldeformation profile in order to achieve a resulting global curvatureprofile that is approximately equal to the desired global curvatureprofile during the carved turn; modifying the sidecut profile based atleast on the torsional deformation profile in order to achieve aresulting snow penetration depth profile that is approximately equal tothe desired snow penetration depth profile; and modifying the widthprofile based at least on the modified sidecut profile; and define thedesign for the gliding board based at least on the width profile, thecamber profile, and the at least one stiffness design variable profile.2. The method of claim 1, wherein modifying at least the width profile,the sidecut profile and at least one of the at least one stiffnessdesign variable profile at least once comprises: iteratively repeatingthe steps of calculating the desired flexural stiffness profile,modifying at least one of the at least one stiffness design variableprofile, calculating the torsional deformation profile, modifying thedesired total curvature profile, modifying the sidecut profile, andmodifying the width profile until a difference between at least one ofthe width profile, the sidecut profile, and at least one of the at leastone stiffness design variable profile associated with a currentiteration and at least one of the width profile, the sidecut profile,and at least one of the at least one stiffness design variable profileassociated with the previous iteration is less than a predeterminedthreshold.
 3. The method of claim 1, wherein defining the desired globalcurvature profile comprises: defining the desired global curvatureprofile to correspond to a carved turn that exhibits an approximatelyconstant turning radius.
 4. The method of claim 1, wherein each of thedesired global curvature profile and the resulting global curvatureprofile embodies a curvilinear form that can be determined by projectinga curvilinear geometry of a corresponding snow trace profile onto anoblique projection plane that is inclined with respect to the plane onwhich the respective snow trace profile exists.
 5. The method of claim1, wherein generating the sidecut profile comprises: defining a desiredgeometry of a snow penetration depth profile, wherein the snowpenetration depth profile represents the penetration depth of thegliding board into snow during the desired carved turn; and generatingthe sidecut profile based at least on the desired global curvatureprofile, the desired geometry of the snow penetration depth profile, andthe nominal edging angle.
 6. The method of claim 5, wherein defining thedesired geometry of the snow penetration depth profile comprises:defining the desired geometry of the snow penetration depth profile toapproximately follow a linear function over a length of an effectivebase region of the gliding board.
 7. The method of claim 1, whereingenerating the camber profile comprises: determining a first assumedtotal curvature profile of the gliding board and a first load profile ofthe gliding board corresponding to a first gliding board edging angle;determining a second assumed total curvature profile of the glidingboard and a second load profile of the gliding board corresponding to asecond gliding board edging angle; and generating the camber profilesuch that it facilitates a gliding board design that is capable ofsatisfying the first assumed total curvature profile with the first loadprofile, and is also capable of satisfying the second assumed totalcurvature profile with the second load profile.
 8. The method of claim1, wherein at least one of the at least one stiffness design variableprofile comprises a core thickness profile of the gliding board.
 9. Themethod of claim 1, wherein calculating the desired flexural stiffnessprofile comprises: calculating the desired flexural stiffness profile ofthe gliding board based at least on the camber profile, the desiredtotal curvature profile, and the total load profile.
 10. The method ofclaim 1, wherein modifying at least one of the at least one stiffnessdesign variable profile comprises: finding the at least one stiffnessdesign variable profile such that the resulting flexural stiffnessprofile is approximately equal to the desired flexural stiffnessprofile, wherein the at least one stiffness design variable profilecomprises a plurality of local stiffness design variable values, whereineach local stiffness design variable value is limited to a predeterminedrange of allowable values.
 11. The method of claim 10, wherein modifyingat least the width profile, the sidecut profile and at least one of theat least one stiffness design variable profile at least once furthercomprises modifying the camber profile at least once by: determining aresulting flexural stiffness profile of the gliding board based at leaston the at least one stiffness design variable profile, the widthprofile, and at least one gliding board material property; determining aresulting flexural curvature profile based at least on the resultingflexural stiffness profile and the total load profile; and modifying thecamber profile based at least on the resulting flexural curvatureprofile and the desired total curvature profile.
 12. The method of claim1, wherein calculating the torsional deformation profile comprises:determining a resulting torsional stiffness profile of the gliding boardbased at least on the at least one stiffness design variable profile,the width profile, and at least one gliding board material property; andgenerating the torsional deformation profile based at least on theresulting torsional stiffness profile and the total load profile. 13.The method of claim 1, wherein the total load profile comprises a snowload profile, and wherein generating the total load profile comprisescalculating the snow load profile by: determining a desired snowpenetration depth profile such that a total magnitude of the resultingsnow load profile is approximately equal to the total magnitude of arelevant component of the athlete load profile, and such that a positionof a centroid of the resulting snow load profile is approximately equalto a position of a centroid of the relevant component of the athleteload profile.
 14. The method of claim 1, wherein the total load profilecomprises the athlete load profile.
 15. The method of claim 1, whereinthe total load profile comprises at least one binding load profile. 16.The method of claim 1, wherein modifying the desired total curvatureprofile comprises: determining a resulting deformed shape of the glidingboard during the carved turn based at least on the desired totalcurvature profile and the torsional deformation profile; determining aposition and an orientation for the resulting deformed shape of thegliding board, wherein the position and orientation correspond to thedesired snow penetration depth profile; determining a resulting snowtrace profile based at least on the resulting deformed shape of thegliding board and the determined position and orientation; determining aresulting global curvature profile based at least on the resulting snowtrace profile; comparing the resulting global curvature profile to thedesired global curvature profile; and modifying the desired totalcurvature profile based at least on the comparison between the resultingglobal curvature profile and the desired global curvature profile. 17.The method of claim 16, wherein determining the position and theorientation for the resulting deformed shape of the gliding boardcomprises: determining at least one anchor point along a length of thegliding board; and determining the position and the orientation for theresulting deformed shape of the gliding board such that, at each of theat least one anchor point, a resulting snow penetration depth isapproximately equal to a corresponding desired snow penetration depthvalue from the desired snow penetration depth profile.
 18. The method ofclaim 1, wherein modifying the sidecut profile comprises: determining aresulting deformed shape of the gliding board during the carved turnbased at least on the desired total curvature profile and the torsionaldeformation profile; determining a position and an orientation for theresulting deformed shape of the gliding board, wherein the position andorientation correspond to the desired snow penetration depth profile;and modifying the sidecut profile such that a resulting snow penetrationdepth profile is approximately equal to the desired snow penetrationdepth profile.
 19. The method of claim 18, wherein determining theposition and the orientation for the resulting deformed shape of thegliding board comprises: determining at least one anchor point along alength of the gliding board; and determining the position and theorientation for the resulting deformed shape of the gliding board suchthat, at each of the at least one anchor point, a resulting snowpenetration depth is approximately equal to a corresponding desired snowpenetration depth value from the desired snow penetration depth profile.20. The method of claim 1, wherein modifying at least the width profile,the sidecut profile and at least one of the at least one stiffnessdesign variable profile at least once further comprises: modifying thetotal load profile based at least on the athlete load profile.